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Tenth Edition 



Systems of Units. Some Important Conversion Factors 

The most important systems of units are shown in the table below. The mks system is also known as 
the International System of Units (abbreviated 57), and the abbreviations sec (instead of s), 
gm (instead of g), and nt (instead of N) are also used. 

System of units 





cgs system 

centimeter (cm) 

gram (g) 

second (s) 


mks system 

meter (m) 

kilogram (kg) 

second (s) 

newton (nt) 

Engineering system 

foot (ft) 


second (s) 

pound (lb) 

1 inch (in.) = 2.540000 cm 1 foot (ft) = 12 in. = 30.480000 cm 

1 yard (yd) = 3 ft = 91.440000 cm 1 statute mile (mi) = 5280 ft = 1.609344 km 

1 nautical mile = 6080 ft = 1.853184 km 

1 acre = 4840 yd 2 = 4046.8564 m 2 1 mi 2 = 640 acres = 2.5899881 km 2 

1 fluid ounce = 1/128 U.S. gallon = 231/128 in. 3 = 29.573730 cm 3 
1 U.S. gallon = 4 quarts (liq) = 8 pints (liq) = 128 fl oz = 3785.4118 cm 3 
1 British Imperial and Canadian gallon = 1.200949 U.S. gallons = 4546.087 cm 3 
1 slug = 14.59390 kg 

1 pound (lb) = 4.448444 nt 1 newton (nt) = 10 5 dynes 

1 British thermal unit (Btu) = 1054.35 joules 1 joule = 10 7 ergs 

1 calorie (cal) = 4.1840 joules 

1 kilowatt-hour (kWh) = 3414.4 Btu = 3.6 ■ 10 6 joules 
1 horsepower (hp) = 2542.48 Btu/h = 178.298 cal/sec = 0.74570 kW 
1 kilowatt (kW) = 1000 watts = 3414.43 Btu/h = 238.662 cal/s 

°F = °C • 1.8 + 32 1° = 60' = 3600" = 0.017453293 radian 

For further details see, for example, D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics. 9th ed., Hoboken, 
N. J: Wiley, 2011. See also AN American National Standard, ASTM/IEEE Standard Metric Practice, Institute of Electrical and 
Electronics Engineers, Inc. (IEEE), 445 Hoes Lane, Piscataway, N. J. 08854, website at 



(cu)' = cu' (c constant) 

J uv' dx = uv — J u'v dx (by parts) 

(m + v)' = u + v' 

r x n+1 

I x n dx = + c (n ¥= 1) 

J n + 1 

(uv)' = u'v + uv' 

f — dx = In Ixl + c 
J x 11 

( u\ U V — uv 

U/ ~ y2 

f e ax dx = - e ax + c 

J a 

du du dy 

— = — ■ — (Chain rule) 

dx dy dx 

J sin x dx = —cos x + c 
J cos x dx = sin x + c 

i— 1 


J tan x dx = —In |cosx| + c 
J cot x dx = In |sin x| + c 



J sec x dx = In |sec x + tan x| + c 

(e ax ) = ae ax 

esc x dx = In |csc x — cot x| + c 

( a x )' = a x In a 

r dx 1 x 

(sin x)' = cos x 

2 2 — arctan + c 

J x + a a a 

(cosx/ = — sinx 

r dx x 

r-n 2 - arcsin + c 

J Vfl 2 - X 2 a 

(tanx) = sec x 

(colx/ = — csc 2 x 

r dx x 

r -, 5 s — arcsinh + c 

J Vx 2 + a 2 a 

(smlix/ = coshx 

r dx x 

r-s K — arccosh + c 

J Vx 2 - a 2 a 

(coshx) — sinhx 

(In x)' = — 


J sin 2 x dx = \x — \ sin 2x + c 
J cos 2 x dx = |x + \ sin 2x + c 

/i n t ^°8a e 

(log a x) = 


J tan 2 x dx = tan x — x + c 
J cot 2 x dx = —cot x — x + c 

(arcsinx)' = , 1 

Vl - x 2 

^ \n x dx = x\n x — x + c 

(arccosx)' = 7===f 

Vl - x 2 

J e ax sin bx dx 

g ax 

= „ „ (a sin bx b cos bx) + c 

a 2 + b 2 

(arctanx)' = ^ ^ 2 

J e m cos bx dx 

(arccotx)' = - - ^ 

e ax 

= (a cos bx + b sin bx) + c 

c 2 + b 2 



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Professor of Mathematics 
Ohio State University 
Columbus, Ohio 

In collaboration with 


New York, New York 


Associate Professor of Mathematics 
Carleton University 
Ottawa, Ontario 




Laurie Rosatone 
Shannon Corliss 
Jonathan Cottrell 
Lucille Buonocore 
Barbara Russiello 
Melissa Edwards 
Lisa Sabatini 

Madelyn Lesure 
Sheena Goldstein 

© Denis Jr. Tangney/iStockphoto 

Cover photo shows the Zakim Bunker Hill Memorial Bridge in 
Boston, MA. 

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ISBN 978-0-470-45836-5 

Printed in the United States of America 

10 987654321 


See also 

Purpose and Structure of the Book 

This book provides a comprehensive, thorough, and up-to-date treatment of engineering 
mathematics . It is intended to introduce students of engineering, physics, mathematics, 
computer science, and related fields to those areas of applied mathematics that are most 
relevant for solving practical problems. A course in elementary calculus is the sole 
prerequisite . (However, a concise refresher of basic calculus for the student is included 
on the inside cover and in Appendix 3.) 

The subject matter is arranged into seven parts as follows: 

A. Ordinary Differential Equations (ODEs) in Chapters 1-6 

B. Linear Algebra. Vector Calculus. See Chapters 7-10 

C. Fourier Analysis. Partial Differential Equations (PDEs). See Chapters 11 and 12 

D. Complex Analysis in Chapters 13-18 

E. Numeric Analysis in Chapters 19-21 
Optimization, Graphs in Chapters 22 and 23 

G. Probability, Statistics in Chapters 24 and 25. 

These are followed by five appendices: 1. References, 2. Answers to Odd-Numbered 
Problems, 3. Auxiliary Materials (see also inside covers of book), 4. Additional Proofs, 
5 Table of Functions. This is shown in a block diagram on the next page. 

The parts of the book are kept independent. In addition, individual chapters are kept as 
independent as possible. (If so needed, any prerequisites — to the level of individual 
sections of prior chapters — are clearly stated at the opening of each chapter.) We give the 
instructor maximum flexibility in selecting the material and tailoring it to his or her 
need. The book has helped to pave the way for the present development of engineering 
mathematics. This new edition will prepare the student for the current tasks and the future 
by a modern approach to the areas listed above. We provide the material and learning 
tools for the students to get a good foundation of engineering mathematics that will help 
them in their careers and in further studies. 

General Features of the Book Include: 

Simplicity of examples to make the book teachable — why choose complicated 
examples when simple ones are as instructive or even better? 

Independence of parts and blocks of chapters to provide flexibility in tailoring 
courses to specific needs. 

Self-contained presentation, except for a few clearly marked places where a proof 
would exceed the level of the book and a reference is given instead. 

Gradual increase in difficulty of material with no jumps or gaps to ensure an 
enjoyable teaching and learning experience. 

Modern standard notation to help students with other courses, modem books, and 
journals in mathematics, engineering, statistics, physics, computer science, and others. 

Furthermore, we designed the book to be a single, self-contained, authoritative, and 
convenient source for studying and teaching applied mathematics, eliminating the need 
for time-consuming searches on the Internet or time-consuming trips to the library to get 
a particular reference book. 






Chaps. 1-6 

Ordinary Differential Equations (ODEs) 


s. 1-4 

Basic Material 

Chap. 5 

Chap. 6 

Series Solutions 

Laplace Transforms 


Chaps. 7-10 

Linear Algebra. 

Vector Calculus 

Chap. 7 

Chap. 9 


Vector Differential 

Linear Systems 




Chap. 8 

Chap. 10 

Eigenvalue Problems 

Vector Integral Calculus 


Chaps. 11-12 

Fourier Analysis. Partial Differential 
Equations (PDEs) 

Chap. 11 

Fourier Analysis 
Chap. 12 

Partial Differential Equations 


Chaps. 13-18 
Complex Analysis, 
Potential Theory 

Chaps. 13-17 
Basic Material 

Chap. 18 

Potential Theory 


Chaps. 19-21 
Numeric Analysis 


Chaps. 22-23 
Optimization, Graphs 

Chap. 19 

Chap. 20 

Chap. 21 

Chap. 22 

Chap. 23 

Numerics in 


Numerics for 

Linear Programming 

Graphs, Optimization 


Linear Algebra 

ODEs and PDEs 


Chaps. 24-25 
Probability, Statistics 

Chap. 24 

Data Analysis. Probability Theory 
Chap. 25 

Mathematical Statistics 


Maple Computer Guide 
Mathematica Computer Guide 

Student Solutions Manual 
and Study Guide 

Instructor’s Manual 



Four Underlying Themes of the Book 

The driving force in engineering mathematics is the rapid growth of technology and the 
sciences. New areas — often drawing from several disciplines — come into existence. 
Electric cars, solar energy, wind energy, green manufacturing, nanotechnology, risk 
management, biotechnology, biomedical engineering, computer vision, robotics, space 
travel, communication systems, green logistics, transportation systems, financial 
engineering, economics, and many other areas are advancing rapidly. What does this mean 
for engineering mathematics? The engineer has to take a problem from any diverse area 
and be able to model it. This leads to the first of four underlying themes of the book. 

1. Modeling is the process in engineering, physics, computer science, biology, 
chemistry, environmental science, economics, and other fields whereby a physical situation 
or some other observation is translated into a mathematical model. This mathematical 
model could be a system of differential equations, such as in population control (Sec. 4.5), 
a probabilistic model (Chap. 24), such as in risk management, a linear programming 
problem (Secs. 22.2-22.4) in minimizing environmental damage due to pollutants, a 
financial problem of valuing a bond leading to an algebraic equation that has to be solved 
by Newton’s method (Sec. 19.2), and many others. 

The next step is solving the mathematical problem obtained by one of the many 
techniques covered in Advanced Engineering Mathematics. 

The third step is interpreting the mathematical result in physical or other terms to 
see what it means in practice and any implications. 

Finally, we may have to make a decision that may be of an industrial nature or 
recommend a public policy. For example, the population control model may imply 
the policy to stop fishing for 3 years. Or the valuation of the bond may lead to a 
recommendation to buy. The variety is endless, but the underlying mathematics is 
surprisingly powerful and able to provide advice leading to the achievement of goals 
toward the betterment of society, for example, by recommending wise policies 
concerning global warming, better allocation of resources in a manufacturing process, 
or making statistical decisions (such as in Sec. 25.4 whether a drug is effective in treating 
a disease). 

While we cannot predict what the future holds, we do know that the student has to 
practice modeling by being given problems from many different applications as is done 
in this book. We teach modeling from scratch, right in Sec. 1.1, and give many examples 
in Sec. 1.3, and continue to reinforce the modeling process throughout the book. 

2. Judicious use of powerful software for numerics (listed in the beginning of Part E) 
and statistics (Part G) is of growing importance. Projects in engineering and industrial 
companies may involve large problems of modeling very complex systems with hundreds 
of thousands of equations or even more. They require the use of such software. However, 
our policy has always been to leave it up to the instructor to determine the degree of use of 
computers, from none or little use to extensive use. More on this below. 

3. The beauty of engineering mathematics. Engineering mathematics relies on 
relatively few basic concepts and involves powerful unifying principles. We point them 
out whenever they are clearly visible, such as in Sec. 4.1 where we “grow” a mixing 
problem from one tank to two tanks and a circuit problem from one circuit to two circuits, 
thereby also increasing the number of ODEs from one ODE to two ODEs. This is an 
example of an attractive mathematical model because the “growth” in the problem is 
reflected by an “increase” in ODEs. 



4. To clearly identify the conceptual structure of subject matters. For example, 
complex analysis (in Part D) is a field that is not monolithic in structure but was formed 
by three distinct schools of mathematics. Each gave a different approach, which we clearly 
mark. The first approach is solving complex integrals by Cauchy’s integral formula (Chaps. 
13 and 14), the second approach is to use the Laurent series and solve complex integrals 
by residue integration (Chaps. 15 and 16), and finally we use a geometric approach of 
conformal mapping to solve boundary value problems (Chaps. 17 and 18). Learning the 
conceptual structure and terminology of the different areas of engineering mathematics is 
very important for three reasons: 

a. It allows the student to identify a new problem and put it into the right group of 
problems. The areas of engineering mathematics are growing but most often retain their 
conceptual structure. 

b. The student can absorb new information more rapidly by being able to fit it into the 
conceptual structure. 

c. Knowledge of the conceptual structure and terminology is also important when using 
the Internet to search for mathematical information. Since the search proceeds by putting 
in key words (i.e., terms) into the search engine, the student has to remember the important 
concepts (or be able to look them up in the book) that identify the application and area 
of engineering mathematics. 

Big Changes in This Edition 

Q Problem Sets Changed 

The problem sets have been revised and rebalanced with some problem sets having more 
problems and some less, reflecting changes in engineering mathematics. There is a greater 
emphasis on modeling. Now there are also problems on the discrete Lourier transform 
(in Sec. 11.9). 

Q Series Solutions of ODEs, Special Functions and Fourier Analysis Reorganized 

Chap. 5, on series solutions of ODEs and special functions, has been shortened. Chap. 1 1 
on Lourier Analysis now contains Sturm-Liouville problems, orthogonal functions, and 
orthogonal eigenfunction expansions (Secs. 1 1.5, 1 1.6), where they fit better conceptually 
(rather than in Chap. 5), being extensions of Fourier’s idea of using orthogonal functions. 

€> Openings of Parts and Chapters Rewritten As Well As Parts of Sections 

In order to give the student a better idea of the structure of the material (see Underlying 
Theme 4 above), we have entirely rewritten the openings of parts and chapters. 
Furthermore, large parts or individual paragraphs of sections have been rewritten or new 
sentences inserted into the text. This should give the students a better intuitive 
understanding of the material (see Theme 3 above), let them draw conclusions on their 
own, and be able to tackle more advanced material. Overall, we feel that the book has 
become more detailed and leisurely written. 

f|H Student Solutions Manual and Study Guide Enlarged 

Upon the explicit request of the users, the answers provided are more detailed and 
complete. More explanations are given on how to learn the material effectively by pointing 
out what is most important. 

Q More Historical Footnotes, Some Enlarged 

Historical footnotes are there to show the student that many people from different countries 
working in different professions, such as surveyors, researchers in industry, etc., contributed 



to the field of engineering mathematics. It should encourage the students to be creative in 
their own interests and careers and perhaps also to make contributions to engineering 

Further Changes and New Features 

Parts of Chap. 1 on first-order ODEs are rewritten. More emphasis on modeling, also 
new block diagram explaining this concept in Sec. 1.1. Early introduction of Euler’s 
method in Sec. 1.2 to familiarize student with basic numerics. More examples of 
separable ODEs in Sec. 1.3. 

For Chap. 2, on second-order ODEs, note the following changes: For ease of reading, 
the first part of Sec. 2.4, which deals with setting up the mass-spring system, has 
been rewritten; also some rewriting in Sec. 2.5 on the Euler-Cauchy equation. 

Substantially shortened Chap. 5, Series Solutions of ODEs. Special Functions: 
combined Secs. 5.1 and 5.2 into one section called “Power Series Method,” shortened 
material in Sec. 5.4 Bessel’s Equation (of the first kind), removed Sec. 5.7 
(Sturm-Liouville Problems) and Sec. 5.8 (Orthogonal Eigenfunction Expansions) and 
moved material into Chap. 1 1 (see “Major Changes” above). 

New equivalent definition of basis (Sec. 7.4). 

In Sec. 7.9, completely new part on composition of linear transformations with 
two new examples. Also, more detailed explanation of the role of axioms, in 
connection with the definition of vector space. 

New table of orientation (opening of Chap. 8 “Linear Algebra: Matrix Eigenvalue 
Problems”) where eigenvalue problems occur in the book. More intuitive explanation 
of what an eigenvalue is at the begining of Sec. 8.1. 

Better definition of cross product (in vector differential calculus) by properly 
identifying the degenerate case (in Sec. 9.3). 

Chap. 11 on Fourier Analysis extensively rearranged: Secs. 11.2 and 11.3 

combined into one section (Sec. 11.2), old Sec. 11.4 on complex Fourier Series 
removed and new Secs. 11.5 (Sturm-Liouville Problems) and 11.6 (Orthogonal 
Series) put in (see “Major Changes” above). New problems (new!) in problem set 

11.9 on discrete Fourier transform. 

New section 12.5 on modeling heat flow from a body in space by setting up the heat 
equation. Modeling PDEs is more difficult so we separated the modeling process 
from the solving process (in Sec. 12.6). 

Introduction to Numerics rewritten for greater clarity and better presentation; new 
Example 1 on how to round a number. Sec. 19.3 on interpolation shortened by 
removing the less important central difference formula and giving a reference instead. 

Large new footnote with historical details in Sec. 22.3, honoring George Dantzig, 
the inventor of the simplex method. 

Traveling salesman problem now described better as a “difficult” problem, typical 
of combinatorial optimization (in Sec. 23.2). More careful explanation on how to 
compute the capacity of a cut set in Sec. 23.6 (Flows on Networks). 

In Chap. 24, material on data representation and characterization restructured in 
terms of five examples and enlarged to include empirical rule on distribution of 



data, outliers, and the score (Sec. 24.1). Furthermore, new example on encription 
(Sec. 24.4). 

Lists of software for numerics (Part E) and statistics (Part G) updated. 

References in Appendix 1 updated to include new editions and some references to 

Use of Computers 

The presentation in this book is adaptable to various degrees of use of software, 
Computer Algebra Systems (CAS’s), or programmable graphic calculators, ranging 
from no use, very little use, medium use, to intensive use of such technology. The choice 
of how much computer content the course should have is left up to the instructor, thereby 
exhibiting our philosophy of maximum flexibility and adaptability. And, no matter what 
the instructor decides, there will be no gaps or jumps in the text or problem set. Some 
problems are clearly designed as routine and drill exercises and should be solved by 
hand (paper and pencil, or typing on your computer). Other problems require more 
thinking and can also be solved without computers. Then there are problems where the 
computer can give the student a hand. And finally, the book has CAS projects, CAS 
problems and CAS experiments , which do require a computer, and show its power in 
solving problems that are difficult or impossible to access otherwise. Here our goal is 
to combine intelligent computer use with high-quality mathematics. The computer 
invites visualization, experimentation, and independent discovery work. In summary, 
the high degree of flexibility of computer use for the book is possible since there are 
plenty of problems to choose from and the CAS problems can be omitted if desired. 

Note that information on software (what is available and where to order it) is at the 
beginning of Part E on Numeric Analysis and Part G on Probability and Statistics. Since 
Maple and Mathematica are popular Computer Algebra Systems, there are two computer 
guides available that are specifically tailored to Advanced Engineering Mathematics: 
E. Kreyszig and E.J. Norminton, Maple Computer Guide, 10th Edition and Mathematica 
Computer Guide. 10 th Edition. Their use is completely optional as the text in the book is 
written without the guides in mind. 

Suggestions for Courses: A Four-Semester Sequence 

The material, when taken in sequence, is suitable for four consecutive semester courses, 
meeting 3 to 4 hours a week: 

1st Semester 
2nd Semester 
3rd Semester 
4th Semester 

ODEs (Chaps. 1-5 or 1-6) 

Linear Algebra. Vector Analysis (Chaps. 7-10) 
Complex Analysis (Chaps. 13-18) 

Numeric Methods (Chaps. 19-21) 

Suggestions for Independent One-Semester Courses 

The book is also suitable for various independent one-semester courses meeting 3 hours 
a week. For instance. 

Introduction to ODEs (Chaps. 1-2, 21.1) 

Laplace Transforms (Chap. 6) 

Matrices and Linear Systems (Chaps. 7-8) 

Vector Algebra and Calculus (Chaps. 9-10) 

Fourier Series and PDEs (Chaps. 11-12, Secs. 21.4—21.7) 

Introduction to Complex Analysis (Chaps. 13-17) 

Numeric Analysis (Chaps. 19, 21) 

Numeric Linear Algebra (Chap. 20) 

Optimization (Chaps. 22-23) 

Graphs and Combinatorial Optimization (Chap. 23) 

Probability and Statistics (Chaps. 24-25) 


We are indebted to former teachers, colleagues, and students who helped us directly or 
indirectly in preparing this book, in particular this new edition. We profited greatly from 
discussions with engineers, physicists, mathematicians, computer scientists, and others, 
and from their written comments. We would like to mention in particular Professors 

Y. A. Antipov, R. Belinski, S. L. Campbell, R. Carr, P. L. Chambre, Isabel F. Cruz, 

Z. Davis, D. Dicker, L. D. Drager, D. Ellis, W. Fox, A. Goriely, R. B. Guenther, 
J. B. Handley, N. Harbertson, A. Hassen, V. W. Howe, H. Kuhn, K. Millet, J. D. Moore, 
W. D. Munroe, A. Nadim, B. S. Ng, J. N. Ong, P. J. Pritchard, W. O. Ray, L. F. Shampine, 
H. L. Smith, Roberto Tamassia, A. L. Villone, H. J. Weiss, A. Wilansky, Neil M. Wigley, 
and L. Ying; Maria E. and Jorge A. Miranda, JD, all from the United States; Professors 
Wayne H. Enright, Francis. L. Lemire, James J. Little, David G. Lowe, Gerry McPhail, 
Theodore S. Norvell, and R. Vaillancourt; Jeff Seiler and David Stanley, all from Canada; 
and Professor Eugen Eichhorn, Gisela Heckler, Dr. Gunnar Schroeder, and Wiltrud 
Stiefenhofer from Europe. Furthermore, we would like to thank Professors John 
B. Donaldson, Bruce C. N. Greenwald, Jonathan L. Gross, Morris B. Holbrook, John 
R. Render, and Bernd Schmitt; and Nicholaiv Villalobos, all from Columbia University, 
New York; as well as Dr. Pearl Chang, Chris Gee, Mike Hale, Joshua Jayasingh, MD, 
David Kahr, Mike Lee, R. Richard Royce, Elaine Schattner, MD, Raheel Siddiqui, Robert 
Sullivan, MD, Nancy Veit, and Ana M. Kreyszig, JD, all from New York City. We would 
also like to gratefully acknowledge the use of facilities at Carleton University, Ottawa, 
and Columbia University, New York. 

Furthermore we wish to thank John Wiley and Sons, in particular Publisher Laurie 
Rosatone, Editor Shannon Corliss, Production Editor Barbara Russiello, Media Editor 
Melissa Edwards, Text and Cover Designer Madelyn Lesure, and Photo Editor Sheena 
Goldstein for their great care and dedication in preparing this edition. In the same vein, 
we would also like to thank Beatrice Ruberto, copy editor and proofreader, WordCo, for 
the Index, and Joyce Franzen of PreMedia and those of PreMedia Global who typeset this 

Suggestions of many readers worldwide were evaluated in preparing this edition. 
Further comments and suggestions for improving the book will be gratefully received. 




Ordinary Differential Equations (ODEs) 1 

CHAPTER 1 First-Order ODEs 2 

1.1 Basic Concepts. Modeling 2 

1.2 Geometric Meaning of y = f(x, y). Direction Fields, Euler’s Method 9 

1.3 Separable ODEs. Modeling 12 

1.4 Exact ODEs. Integrating Factors 20 

1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 27 

1.6 Orthogonal Trajectories. Optional 36 

1.7 Existence and Uniqueness of Solutions for Initial Value Problems 38 
Chapter 1 Review Questions and Problems 43 

Summary of Chapter 1 44 

CHAPTER 2 Second-Order Linear ODEs 46 

2.1 Homogeneous Linear ODEs of Second Order 46 

2.2 Homogeneous Linear ODEs with Constant Coefficients 53 

2.3 Differential Operators. Optional 60 

2.4 Modeling of Free Oscillations of a Mass-Spring System 62 

2.5 Euler-Cauchy Equations 71 

2.6 Existence and Uniqueness of Solutions. Wronskian 74 

2.7 Nonhomogeneous ODEs 79 

2.8 Modeling: Forced Oscillations. Resonance 85 

2.9 Modeling: Electric Circuits 93 

2.10 Solution by Variation of Parameters 99 
Chapter 2 Review Questions and Problems 102 
Summary of Chapter 2 103 

CHAPTER 3 Higher Order Linear ODEs 105 

3.1 Homogeneous Linear ODEs 105 

3.2 Homogeneous Linear ODEs with Constant Coefficients 111 

3.3 Nonhomogeneous Linear ODEs 116 
Chapter 3 Review Questions and Problems 122 
Summary of Chapter 3 123 

CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods 124 

4.0 For Reference: Basics of Matrices and Vectors 124 

4.1 Systems of ODEs as Models in Engineering Applications 130 

4.2 Basic Theory of Systems of ODEs. Wronskian 137 

4.3 Constant-Coefficient Systems. Phase Plane Method 140 

4.4 Criteria for Critical Points. Stability 148 

4.5 Qualitative Methods for Nonlinear Systems 152 

4.6 Nonhomogeneous Linear Systems of ODEs 160 
Chapter 4 Review Questions and Problems 164 
Summary of Chapter 4 165 

CHAPTER 5 Series Solutions of ODEs. Special Functions 167 

5.1 Power Series Method 167 

5.2 Legendre’s Equation. Legendre Polynomials P n (x ) 175 




5.3 Extended Power Series Method: Frobenius Method 180 

5.4 Bessel’s Equation. Bessel Functions J v (x ) 187 

5.5 Bessel Functions of the Y v (x). General Solution 196 
Chapter 5 Review Questions and Problems 200 
Summary of Chapter 5 201 

CHAPTER 6 Laplace Transforms 203 

6.1 Laplace Transform. Linearity. First Shifting Theorem (^-Shifting) 204 

6.2 Transforms of Derivatives and Integrals. ODEs 211 

6.3 Unit Step Function (Heaviside Function). 

Second Shifting Theorem (r-Shifting) 217 

6.4 Short Impulses. Dirac’s Delta Function. Partial Fractions 225 

6.5 Convolution. Integral Equations 232 

6.6 Differentiation and Integration of Transforms. 

ODEs with Variable Coefficients 238 

6.7 Systems of ODEs 242 

6.8 Laplace Transform: General Formulas 248 

6.9 Table of Laplace Transforms 249 
Chapter 6 Review Questions and Problems 251 
Summary of Chapter 6 253 

PART B Linear Algebra. Vector Calculus 255 

CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. 

Linear Systems 256 

7.1 Matrices, Vectors: Addition and Scalar Multiplication 257 

7.2 Matrix Multiplication 263 

7.3 Linear Systems of Equations. Gauss Elimination 272 

7.4 Linear Independence. Rank of a Matrix. Vector Space 282 

7.5 Solutions of Linear Systems: Existence, Uniqueness 288 

7.6 For Reference: Second- and Third-Order Determinants 291 

7.7 Determinants. Cramer’s Rule 293 

7.8 Inverse of a Matrix. Gauss-Jordan Elimination 301 

7.9 Vector Spaces, Inner Product Spaces. Linear Transformations. Optional 309 
Chapter 7 Review Questions and Problems 318 

Summary of Chapter 7 320 

CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems 322 

8.1 The Matrix Eigenvalue Problem. 

Determining Eigenvalues and Eigenvectors 323 

8.2 Some Applications of Eigenvalue Problems 329 

8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices 334 

8.4 Eigenbases. Diagonalization. Quadratic Forms 339 

8.5 Complex Matrices and Forms. Optional 346 
Chapter 8 Review Questions and Problems 352 
Summary of Chapter 8 353 



CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl 354 

9.1 Vectors in 2-Space and 3-Space 354 

9.2 Inner Product (Dot Product) 361 

9.3 Vector Product (Cross Product) 368 

9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives 375 

9.5 Curves. Arc Length. Curvature. Torsion 381 

9.6 Calculus Review: Functions of Several Variables. Optional 392 

9.7 Gradient of a Scalar Field. Directional Derivative 395 

9.8 Divergence of a Vector Field 402 

9.9 Curl of a Vector Field 406 

Chapter 9 Review Questions and Problems 409 
Summary of Chapter 9 410 

CHAPTER 10 Vector Integral Calculus. Integral Theorems 413 

10.1 Line Integrals 413 

10.2 Path Independence of Line Integrals 419 

10.3 Calculus Review: Double Integrals. Optional 426 

10.4 Green’s Theorem in the Plane 433 

10.5 Surfaces for Surface Integrals 439 

10.6 Surface Integrals 443 

10.7 Triple Integrals. Divergence Theorem of Gauss 452 

10.8 Further Applications of the Divergence Theorem 458 

10.9 Stokes’s Theorem 463 

Chapter 10 Review Questions and Problems 469 
Summary of Chapter 10 470 

PART C Fourier Analysis. Partial Differential Equations (PDEs) 473 

CHAPTER 11 Fourier Analysis 474 

11.1 Fourier Series 474 

11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions 483 

11.3 Forced Oscillations 492 

11.4 Approximation by Trigonometric Polynomials 495 

11.5 Sturm-Liouville Problems. Orthogonal Functions 498 

11.6 Orthogonal Series. Generalized Fourier Series 504 

11.7 Fourier Integral 510 

11.8 Fourier Cosine and Sine Transforms 518 

11.9 Fourier Transform. Discrete and Fast Fourier Transforms 522 

11.10 Tables of Transforms 534 

Chapter 1 1 Review Questions and Problems 537 
Summary of Chapter 1 1 538 

CHAPTER 12 Partial Differential Equations (PDEs) 540 

12.1 Basic Concepts of PDEs 540 

12.2 Modeling: Vibrating String, Wave Equation 543 

12.3 Solution by Separating Variables. Use of Fourier Series 545 

12.4 D’Alembert’s Solution of the Wave Equation. Characteristics 553 

12.5 Modeling: Heat Flow from a Body in Space. Heat Equation 557 



12.6 Heat Equation: Solution by Fourier Series. 

Steady Two-Dimensional Heat Problems. Dirichlet Problem 558 

12.7 Heat Equation: Modeling Very Long Bars. 

Solution by Fourier Integrals and Transforms 568 

12.8 Modeling: Membrane, Two-Dimensional Wave Equation 575 

12.9 Rectangular Membrane. Double Fourier Series 577 

12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier-Bessel Series 585 

12.11 Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential 593 

12.12 Solution of PDEs by Laplace Transforms 600 
Chapter 12 Review Questions and Problems 603 
Summary of Chapter 12 604 

PART D Complex Analysis 607 

CHAPTER 13 Complex Numbers and Functions. 

Complex Differentiation 608 

13.1 Complex Numbers and Their Geometric Representation 608 

13.2 Polar Form of Complex Numbers. Powers and Roots 613 

13.3 Derivative. Analytic Function 619 

13.4 Cauchy-Riemann Equations. Laplace’s Equation 625 

13.5 Exponential Function 630 

13.6 Trigonometric and Hyperbolic Functions. Euler’s Formula 633 

13.7 Logarithm. General Power. Principal Value 636 
Chapter 13 Review Questions and Problems 641 
Summary of Chapter 13 641 

CHAPTER 14 Complex Integration 643 

14.1 Line Integral in the Complex Plane 643 

14.2 Cauchy’s Integral Theorem 652 

14.3 Cauchy’s Integral Formula 660 

14.4 Derivatives of Analytic Functions 664 
Chapter 14 Review Questions and Problems 668 
Summary of Chapter 14 669 

CHAPTER 15 Power Series, Taylor Series 671 

15.1 Sequences, Series, Convergence Tests 671 

15.2 Power Series 680 

15.3 Functions Given by Power Series 685 

15.4 Taylor and Maclaurin Series 690 

15.5 Uniform Convergence. Optional 698 
Chapter 15 Review Questions and Problems 706 
Summary of Chapter 15 706 

CHAPTER 16 Laurent Series. Residue Integration 708 

16.1 Laurent Series 708 

16.2 Singularities and Zeros. Infinity 715 

16.3 Residue Integration Method 719 

16.4 Residue Integration of Real Integrals 725 
Chapter 16 Review Questions and Problems 733 
Summary of Chapter 16 734 



CHAPTER 17 Conformal Mapping 736 

17.1 Geometry of Analytic Functions: Conformal Mapping 737 

17.2 Linear Fractional Transformations (Mobius Transformations) 742 

17.3 Special Linear Fractional Transformations 746 

17.4 Conformal Mapping by Other Functions 750 

17.5 Riemann Surfaces. Optional 754 
Chapter 17 Review Questions and Problems 756 
Summary of Chapter 17 757 

CHAPTER 18 Complex Analysis and Potential Theory 758 

18.1 Electrostatic Fields 759 

18.2 Use of Conformal Mapping. Modeling 763 

18.3 Heat Problems 767 

18.4 Fluid Flow 771 

18.5 Poisson’s Integral Formula for Potentials 777 

18.6 General Properties of Harmonic Functions. 

Uniqueness Theorem for the Dirichlet Problem 781 

Chapter 18 Review Questions and Problems 785 
Summary of Chapter 18 786 

PART E Numeric Analysis 787 

Software 788 

CHAPTER 19 Numerics in General 790 

19.1 Introduction 790 

19.2 Solution of Equations by Iteration 798 

19.3 Interpolation 808 

19.4 Spline Interpolation 820 

19.5 Numeric Integration and Differentiation 827 
Chapter 19 Review Questions and Problems 841 
Summary of Chapter 19 842 

CHAPTER 20 Numeric Linear Algebra 844 

20.1 Linear Systems: Gauss Elimination 844 

20.2 Linear Systems: LU-Factorization, Matrix Inversion 852 

20.3 Linear Systems: Solution by Iteration 858 

20.4 Linear Systems: Ill-Conditioning, Norms 864 

20.5 Least Squares Method 872 

20.6 Matrix Eigenvalue Problems: Introduction 876 

20.7 Inclusion of Matrix Eigenvalues 879 

20.8 Power Method for Eigenvalues 885 

20.9 Tridiagonalization and QR-Factorization 888 
Chapter 20 Review Questions and Problems 896 
Summary of Chapter 20 898 

CHAPTER 21 Numerics for ODEs and PDEs 900 

21. Methods for First-Order ODEs 901 

21.2 Multistep Methods 911 

21.3 Methods for Systems and Higher Order ODEs 915 



21.4 Methods for Elliptic PDEs 922 

21.5 Neumann and Mixed Problems. Irregular Boundary 931 

21.6 Methods for Parabolic PDEs 936 

21.7 Method for Hyperbolic PDEs 942 
Chapter 21 Review Questions and Problems 945 
Summary of Chapter 21 946 

PART F Optimization, Graphs 949 

CHAPTER 22 Unconstrained Optimization. Linear Programming 950 

22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent 951 

22.2 Linear Programming 954 

22.3 Simplex Method 958 

22.4 Simplex Method: Difficulties 962 
Chapter 22 Review Questions and Problems 968 
Summary of Chapter 22 969 

CHAPTER 23 Graphs. Combinatorial Optimization 970 

23.1 Graphs and Digraphs 970 

23.2 Shortest Path Problems. Complexity 975 

23.3 Bellman’s Principle. Dijkstra’s Algorithm 980 

23.4 Shortest Spanning Trees: Greedy Algorithm 984 

23.5 Shortest Spanning Trees: Prim’s Algorithm 988 

23.6 Flows in Networks 991 

23.7 Maximum Flow: Ford-Fulkerson Algorithm 998 

23.8 Bipartite Graphs. Assignment Problems 1001 
Chapter 23 Review Questions and Problems 1006 
Summary of Chapter 23 1007 

PART G Probability, Statistics 1009 

Software 1009 

CHAPTER 24 Data Analysis. Probability Theory 1011 

24.1 Data Representation. Average. Spread 1011 

24.2 Experiments, Outcomes, Events 1015 

24.3 Probability 1018 

24.4 Permutations and Combinations 1024 

24.5 Random Variables. Probability Distributions 1029 

24.6 Mean and Variance of a Distribution 1035 

24.7 Binomial, Poisson, and Hypergeometric Distributions 1039 

24.8 Normal Distribution 1045 

24.9 Distributions of Several Random Variables 1051 
Chapter 24 Review Questions and Problems 1060 
Summary of Chapter 24 1060 

CHAPTER 25 Mathematical Statistics 1063 

25.1 Introduction. Random Sampling 1063 

25.2 Point Estimation of Parameters 1065 

25.3 Confidence Intervals 1068 


25.4 Testing Hypotheses. Decisions 1077 

25.5 Quality Control 1087 

25.6 Acceptance Sampling 1092 

25.7 Goodness of Fit. ^ 2 -Test 1096 

25.8 Nonparametric Tests 1100 

25.9 Regression. Fitting Straight Lines. Correlation 1103 
Chapter 25 Review Questions and Problems 1111 
Summary of Chapter 25 1112 

APPENDIX 1 References A1 

APPENDIX 2 Answers to Odd-Numbered Problems 

APPENDIX 3 Auxiliary Material A63 

A3.1 Formulas for Special Functions A63 
A3.2 Partial Derivatives A69 
A3.3 Sequences and Series A72 

A3.4 Grad, Div, Curl, V 2 in Curvilinear Coordinates A74 

APPENDIX 4 Additional Proofs A77 

APPENDIX 5 Tables A97 



Equations (ODEs) 


First-Order ODEs 
Second-Order Linear ODEs 
Higher Order Linear ODEs 

Systems of ODEs. Phase Plane. Qualitative Methods 
Series Solutions of ODEs. Special Functions 
Laplace Transforms 

Many physical laws and relations can be expressed mathematically in the form of differential 
equations. Thus it is natural that this book opens with the study of differential equations and 
their solutions. Indeed, many engineering problems appear as differential equations. 

The main objectives of Part A are twofold: the study of ordinary differential equations 
and their most important methods for solving them and the study of modeling. 

Ordinary differential equations (ODEs) are differential equations that depend on a single 
variable. The more difficult study of partial differential equations (PDEs), that is, 
differential equations that depend on several variables, is covered in Part C. 

Modeling is a crucial general process in engineering, physics, computer science, biology, 
medicine, environmental science, chemistry, economics, and other fields that translates a 
physical situation or some other observations into a “mathematical model.” Numerous 
examples from engineering (e.g., mixing problem), physics (e.g., Newton’s law of cooling), 
biology (e.g., Gompertz model), chemistry (e.g., radiocarbon dating), environmental science 
(e.g., population control), etc. shall be given, whereby this process is explained in detail, 
that is, how to set up the problems correctly in terms of differential equations. 

For those interested in solving ODEs numerically on the computer, look at Secs. 21.1-21.3 
of Chapter 21 of Part F, that is, numeric methods for ODEs. These sections are kept 
independent by design of the other sections on numerics. This allows for the study of 
numerics for ODEs directly after Chap. 1 or 2. 











Fig. 1 Modeling, 
solving, interpreting 


First-Order ODEs 

Chapter 1 begins the study of ordinary differential equations (ODEs) by deriving them from 
physical or other problems (modeling), solving them by standard mathematical methods, 
and interpreting solutions and their graphs in terms of a given problem. The simplest ODEs 
to be discussed are ODEs of the first order because they involve only the first derivative 
of the unknown function and no higher derivatives. These unknown functions will usually 
be denoted by y(x) or y(t) when the independent variable denotes time t. The chapter ends 
with a study of the existence and uniqueness of solutions of ODEs in Sec. 1.7. 

Understanding the basics of ODEs requires solving problems by hand (paper and pencil, 
or typing on your computer, but first without the aid of a CAS). In doing so, you will 
gain an important conceptual understanding and feel for the basic terms, such as ODEs, 
direction field, and initial value problem. If you wish, you can use your Computer Algebra 
System (CAS) for checking solutions. 

COMMENT. Numerics for first-order ODEs can be studied immediately after this 
chapter. See Secs. 21.1-21.2, which are independent of other sections on numerics. 

Prerequisite: Integral calculus. 

Sections that may be omitted in a shorter course: 1.6, 1.7. 

References and Answers to Problems: App. 1 Part A, and App. 2. 

Concepts. Modeling 

If we want to solve an engineering problem (usually of a physical nature), we first 
have to formulate the problem as a mathematical expression in terms of variables, 
functions, and equations. Such an expression is known as a mathematical model of the 
given problem. The process of setting up a model, solving it mathematically, and 
interpreting the result in physical or other terms is called mathematical modeling or, 
briefly, modeling. 

Modeling needs experience, which we shall gain by discussing various examples and 
problems. (Your computer may often help you in solving but rarely in setting up models.) 

Now many physical concepts, such as velocity and acceleration, are derivatives. Hence 
a model is very often an equation containing derivatives of an unknown function. Such 
a model is called a differential equation. Of course, we then want to find a solution (a 
function that satisfies the equation), explore its properties, graph it, find values of it, and 
interpret it in physical terms so that we can understand the behavior of the physical system 
in our given problem. However, before we can turn to methods of solution, we must first 
define some basic concepts needed throughout this chapter. 


SEC. 1.1 Basic Concepts. Modeling 


An ordinary differential equation (ODE) is an equation that contains one or several 
derivatives of an unknown function, which we usually call y(x ) (or sometimes y(t) if the 
independent variable is time t). The equation may also contain y itself, known functions 
of x (or t ), and constants. For example, 


y = cos x 


ft . r\ —2X 

y +9 y = e 

( 3 ) 

t tft 3 '2 n 

y y - s y = 0 


CHAP. 1 First-Order ODEs 


are ordinary differential equations (ODEs). Here, as in calculus, y denotes dy/dx, 
y" = d 2 y/dx 2 , etc. The term ordinary distinguishes them from partial differential 
equations (PDEs), which involve partial derivatives of an unknown function of two 
or more variables. For instance, a PDE with unknown function u of two variables x 
and y is 

d 2 u 

dx 2 


d 2 u 

dy 2 

= 0 . 

PDEs have important engineering applications, but they are more complicated than ODEs; 
they will be considered in Chap. 12. 

An ODE is said to be of order n if the nth derivative of the unknown function y is the 
highest derivative of y in the equation. The concept of order gives a useful classification 
into ODEs of first order, second order, and so on. Thus, (1) is of first order, (2) of second 
order, and (3) of third order. 

In this chapter we shall consider first-order ODEs. Such equations contain only the 
first derivative y' and may contain y and any given functions of x. Hence we can write 
them as 


F(x,y,y') = 0 

or often in the form 

/ =f(x,y). 

This is called the explicit form, in contrast to the implicit form (4). For instance, the implicit 
ODE x~ 3 y' — 4 y 2 = 0 (where x A 0) can be written explicitly as y' = 4x 3 y 2 . 

Concept of Solution 

A function 

y = h(x) 

is called a solution of a given ODE (4) on some open interval a < x < b if h(x) is 
defined and differentiable throughout the interval and is such that the equation becomes 
an identity if y and y are replaced with h and h , respectively. The curve (the graph) of 
h is called a solution curve. 

Here, open interval a < x < b means that the endpoints a and b are not regarded as 
points belonging to the interval. Also, a < x < b includes infinite intervals — oo < x < b, 
a < x < °°, — co<^<oo (the real line) as special cases. 

Verification of Solution 

Verify that y = c/x(c an arbitrary constant) is a solution of the ODE xy = — y for all x A 0. Indeed, differentiate 
y = c/x to get y = — c/x . Multiply this by x, obtaining xy’ = — c/x; thus, xy’ = —y, the given ODE. 

SEC. 1.1 Basic Concepts. Modeling 


Solution by Calculus. Solution Curves 

The ODE y = dy/dx = cos* can be solved directly by integration on both sides. Indeed, using calculus, 
we obtain y = f cos * dx = sin * + c, where c is an arbitrary constant. This is a family of solutions . Each value 
of c, for instance, 2.75 or 0 or —8, gives one of these curves. Figure 3 shows some of them, for c = —3, —2, 
-1,0, 1,2, 3,4. ■ 

EXAMPLE 3 (A) Exponential Growth. (B) Exponential Decay 

From calculus we know that y = ce°' 2t has the derivative 

y = % = 0.2e°' 2t = 0.2y. 

Hence y is a solution of y = 0.2y (Fig. 4A). This ODE is of the form y = ky. With positive-constant k it can 
model exponential growth, for instance, of colonies of bacteria or populations of animals. It also applies to 
humans for small populations in a large country (e.g., the United States in early times) and is then known as 
Malthus’s law. 1 We shall say more about this topic in Sec. 1.5. 

(B) Similarly, y = —0.2 (with a minus on the right) has the solution y = ce~°' 2t , (Fig. 4B) modeling 
exponential decay, as, for instance, of a radioactive substance (see Example 5). 


2.5 - 
2.0 - 

1.5 - \ \ 

1.0 - 

0.5 ^ " ' -A 

0 2 4 6 8 10 12 14 t 

Fig. 4B, Solutions of y' = — 0.2y 
in Example 3 (exponential decay) 

Fig. 4 A. Solutions of y' = 0.2y 
in Example 3 (exponential growth) 

1 Named after the English pioneer in classic economics, THOMAS ROBERT MALTHUS (1766-1834). 


CHAP. 1 First-Order ODEs 


We see that each ODE in these examples has a solution that contains an arbitrary 
constant c. Such a solution containing an arbitrary constant c is called a general solution 
of the ODE. 

(We shall see that c is sometimes not completely arbitrary but must be restricted to some 
interval to avoid complex expressions in the solution.) 

We shall develop methods that will give general solutions uniquely (perhaps except for 
notation). Hence we shall say the general solution of a given ODE (instead of a general 

Geometrically, the general solution of an ODE is a family of infinitely many solution 
curves, one for each value of the constant c. If we choose a specific c (e.g., c = 6.45 or 0 
or —2.01) we obtain what is called a particular solution of the ODE. A particular solution 
does not contain any arbitrary constants. 

In most cases, general solutions exist, and every solution not containing an arbitrary 
constant is obtained as a particular solution by assigning a suitable value to c. Exceptions 
to these rules occur but are of minor interest in applications; see Prob. 16 in Problem 
Set 1.1. 

Initial Value Problem 

In most cases the unique solution of a given problem, hence a particular solution, is 
obtained from a general solution by an initial condition y(xo) = yo, with given values 
xq and yo> that is used to determine a value of the arbitrary constant c. Geometrically 
this condition means that the solution curve should pass through the point (xo, yo) 
in the xy-plane. An ODE, together with an initial condition, is called an initial value 
problem. Thus, if the ODE is explicit, y = /(x, v), the initial value problem is of 
the form 

(5) y = fix, y), y(x 0 ) = V 0 . 

Initial Value Problem 

Solve the initial value problem 

, dy 

y = -T- = 3 v, y{ 0) = 5.7. 


Solution. The general solution is ;y(x) = ce 3x \ see Example 3. From this solution and the initial condition 
we obtain y(0) = ce° = c = 5.7. Hence the initial value problem has the solution y(x ) = 5.7e . This is a 
particular solution. 

More on Modeling 

The general importance of modeling to the engineer and physicist was emphasized at the 
beginning of this section. We shall now consider a basic physical problem that will show 
the details of the typical steps of modeling. Step 1 : the transition from the physical situation 
(the physical system) to its mathematical formulation (its mathematical model); Step 2: 
the solution by a mathematical method; and Step 3: the physical interpretation of the result. 
This may be the easiest way to obtain a first idea of the nature and purpose of differential 
equations and their applications. Realize at the outset that your computer (your CAS ) 
may perhaps give you a hand in Step 2, but Steps 1 and 3 are basically your work. 

SEC. 1.1 Basic Concepts. Modeling 


And Step 2 requires a solid knowledge and good understanding of solution methods 
available to you — you have to choose the method for your work by hand or by the 
computer. Keep this in mind, and always check computer results for errors (which may 
arise, for instance, from false inputs). 

EXAMPLE 5 Radioactivity. Exponential Decay 

Given an amount of a radioactive substance, say, 0.5 g (gram), find the amount present at any later time. 

Physical Information. Experiments show that at each instant a radioactive substance decomposes — and is thus 
decaying in time — proportional to the amount of substance present. 

Step 1. Setting up a mathematical model of the physical process. Denote by y(t) the amount of substance still 
present at any time t. By the physical law, the time rate of change y ( t ) = dy/dt is proportional to y(r). This 
gives the first-order ODE 


(6) -=-ky 

where the constant k is positive, so that, because of the minus, we do get decay (as in [B] of Example 3). 
The value of k is known from experiments for various radioactive substances (e.g., k = 1.4 • 10 -11 sec -1 , 
approximately, for radium gsR 3 )- 

Now the given initial amount is 0.5 g, and we can call the corresponding instant t = 0. Then we have the 
initial condition y(0) = 0.5. This is the instant at which our observation of the process begins. It motivates 
the term initial condition (which, however, is also used when the independent variable is not time or when 
we choose a t other than t = 0). Hence the mathematical model of the physical process is the initial value 


(7) — = ~ky, y(0) = 0.5. 


Step 2. Mathematical solution. As in (B) of Example 3 we conclude that the ODE (6) models exponential decay 
and has the general solution (with arbitrary constant c but definite given k) 

(8) y(t) = ce~ kt . 

We now determine c by using the initial condition. Since y(0) = c from (8), this gives y(0) = c = 0.5. Hence 
the particular solution governing our process is (cf. Fig. 5) 

(9) y(t) = 0.5e _fct (k > 0). 

Always check your result — it may involve human or computer errors! Verify by differentiation (chain rule!) 
that your solution (9) satisfies (7) as well as y(0) = 0.5: 

f = -0.5 ke~ kt = -k ■ 0.5e~ kt = - ky , y(0) = 0.5e° = 0.5. 


Step 3. Interpretation of result. Formula (9) gives the amount of radioactive substance at time t. It starts from 
the correct initial amount and decreases with time because k is positive. The limit of y as t — > oo is zero. 

Fig. 5. Radioactivity (Exponential decay, 
y = 0.5e kt , with k = 1.5 as an example) 


CHAP. 1 First-Order ODEs 

PRQB4.ErM=SFT— 1^1 



Solve the ODE by integration or by remembering a 
differentiation formula. 

1. y + 2 sin 2ttx = 0 

2. y + jre _x2/2 = 0 

3- y' = y 

4. y = — 1.5y 

5. y = 4e~ x cos x 

6 n 

■ y = ~y 

7. y = cosh 5. 13 jc 
8. y "' = e~°' 2x 




(a) Verify that y is a solution of the ODE. (b) Determine 

from y the particular solution of the IVP. (c) Graph the 

solution of the IVP. 

9. y' + 4v = 1.4, y = ce~ 4x + 0.35, y(0) = 2 

10. y + 5xy = 0, y = ce~ 25x , v(0) = tt 

11 . y = y + e x , y = (x + c)e x , y(0) = \ 

12. yy = 4x, y 2 - 4x 2 = c(y > 0), y(l) = 4 

13. y'=y- y 2 , y = 1 _ x , y(0) = 0.25 

1 + ce 

14. y' tan* = 2y — 8, y = csin 2 jc + 4, y(^Tr) = 0 

15. Find two constant solutions of the ODE in Prob. 13 by 

16. Singular solution. An ODE may sometimes have an 
additional solution that cannot be obtained from the 
general solution and is then called a singular solution. 
The ODE y' 2 — xy' + y = 0 is of this kind. Show 
by differentiation and substitution that it has the 
general solution y = cx — c 2 and the singular solution 
y = x 2 /4. Explain Fig. 6. 

Fig. 6 



These problems will give you a first impression of modeling. 
Many more problems on modeling follow throughout this 

17. Half-life. The half-life measures exponential decay. 
It is the time in which half of the given amount of 
radioactive substance will disappear. What is the half- 
life of 2 lsRa (in years) in Example 5? 

18. Half-life. Radium 2 l|Ra has a half-life of about 
3.6 days. 

(a) Given 1 gram, how much will still be present after 
1 day? 

(b) After 1 year? 

19. Free fall. In dropping a stone or an iron ball, air 
resistance is practically negligible. Experiments 
show that the acceleration of the motion is constant 
(equal to g = 9.80 m/sec 2 = 32 ft/sec 2 , called the 
acceleration of gravity). Model this as an ODE for 
y(t), the distance fallen as a function of time t. If the 
motion starts at time t = 0 from rest (i.e., with velocity 
v = y' = 0), show that you obtain the familiar law of 
free fall 

y = Jgt 


20. Exponential decay. Subsonic flight. The efficiency 
of the engines of subsonic airplanes depends on air 
pressure and is usually maximum near 35,000 ft. 
Find the air pressure y{x) at this height. Physical 
information. The rate of change y (x) is proportional 
to the pressure. At 18,000 ft it is half its value 
yo = >’(0) at sea level. Hint. Remember from calculus 
that if y = e kx , then y' = ke kx = ky. Can you see 
without calculation that the answer should be close 
to y 0 /4? 

Particular solutions and singular 
solution in Problem 16 

SEC. 1.2 Geometric Meaning of y' = f(x, y). Direction Fields, Euler’s Method 


U Geometric Meaning of y = f(x, y). 
Direction Fields, Eulers Method 

A first-order ODE 

( 1 ) 

y =f(x,y ) 

has a simple geometric interpretation. From calculus you know that the derivative y' (x) of 

y(x) is the slope of y(jc). Hence a solution curve of (1) that passes through a point (jc 0 , y 0 ) 
must have, at that point, the slope y (jc 0 ) equal to the value of/ at that point; that is, 

Using this fact, we can develop graphic or numeric methods for obtaining approximate 
solutions of ODEs (1). This will lead to a better conceptual understanding of an ODE (1). 
Moreover, such methods are of practical importance since many ODEs have complicated 
solution formulas or no solution formulas at all, whereby numeric methods are needed. 

Graphic Method of Direction Fields. Practical Example Illustrated in Fig. 7. We 

can show directions of solution curves of a given ODE (1) by drawing short straight-line 
segments (lineal elements) in the xy-plane. This gives a direction field (or slope field) 
into which you can then fit (approximate) solution curves. This may reveal typical 
properties of the whole family of solutions. 

Figure 7 shows a direction field for the ODE 

obtained by a CAS (Computer Algebra System) and some approximate solution curves 
fitted in. 

/(*o) = f(x(h >’o)- 

( 2 ) 

y = y + x 

\\\\\\\\ \ \-2 \ v 

Fig. 7. Direction field of y' = y + x, with three approximate solution 
curves passing through (0, 1), (0, 0), (0, —1), respectively 


CHAP. 1 First-Order ODEs 

If you have no CAS, first draw a few level curves f(x, y ) = const of f(x, y), then parallel 
lineal elements along each such curve (which is also called an isocline, meaning a curve 
of equal inclination), and finally draw approximation curves fit to the lineal elements. 

We shall now illustrate how numeric methods work by applying the simplest numeric 
method, that is Euler’s method, to an initial value problem involving ODE (2). First we 
give a brief description of Euler’s method. 

Numeric Method by Euler 

Given an ODE (1) and an initial value vGo) = >’o, Euler’s method yields approximate 
solution values at equidistant x- values jt 0 , x\ = xq + h, x 2 = x'o + 2 h, • • ■ , namely, 

yi = Jo + hf(x o, y 0 ) (Fig. 8) 

J 2 = yi + hf(x i,yi), etc. 

In general. 

y n = y n - 1 + ¥(xn-i,y n -i ) 

where the step h equals, e.g., 0.1 or 0.2 (as in Table 1.1) or a smaller value for greater 

Fig. 8. First Euler step, showing a solution curve, its tangent at (x 0 , y 0 ), 
step h and increment hf(x 0 , y 0 ) in the formula for y-| 

Table 1.1 shows the computation of n = 5 steps with step h = 0.2 for the ODE (2) and 
initial condition y(0) = 0, corresponding to the middle curve in the direction field. We 
shall solve the ODE exactly in Sec. 1.5. For the time being, verify that the initial value 
problem has the solution y = e x — x — 1. The solution curve and the values in Table 1.1 
are shown in Fig. 9. These values are rather inaccurate. The errors y(x n ) — y n are shown 
in Table 1.1 as well as in Fig. 9. Decreasing h would improve the values, but would soon 
require an impractical amount of computation. Much better methods of a similar nature 
will be discussed in Sec. 21.1. 

SEC. 1.2 Geometric Meaning of y' = f[x, y). Direction Fields, Euler’s Method 


Table 1." Euler method fory’ = y + x,y(0) = 0 for 
x = 0, , 1.0 with step h — 0.2 


x n 

y n 

y(x n ) 

































Fig. 9. Euler method: Approximate values in Table 1.1 and solution curve 

P^ROBL=E- M= S ^ T ~ l^Z 



Graph a direction field (by a CAS or by hand). In the field 
graph several solution curves by hand, particularly those 
passing through the given points ( x , y). 



= l + 

y 2 , 

(ttt, 1) 



’ + Ax 

= o, 

(1,1), (0, 2) 




= l - 

v 2 , 

(0, 0), (2, i) 




= 2y- 

- y 2 . 

(0, 0), (0, 1), (0, 2), (0, 3) 




= X — 






= sin 2 


(0, -0.4), (0, 1) 




= e v ' x , 

(2, 2), (3, 3) 




= —2xy, 

(0, |), (0, 1), (0, 2) 



Direction fields are very useful because they can give you 
an impression of all solutions without solving the ODE, 
which may be difficult or even impossible. To get a feel for 
the accuracy of the method, graph a field, sketch solution 
curves in it, and compare them with the exact solutions. 

9. y = cos ttx 

10. y = —5y^ 2 (Sol. Vy + f x = c) 

11. Autonomous ODE. This means an ODE not showing 
x (the independent variable) explicitly. (The ODEs in 
Probs. 6 and 10 are autonomous.) What will the level 
curves /(jc, y) = const (also called isoclines = curves 

of equal inclination) of an autonomous ODE look like? 
Give reason. 



Model the motion of a body B on a straight line with 
velocity as given, y (f) being the distance of B from a point 
y = 0 at time t. Graph a direction field of the model (the 
ODE). In the field sketch the solution curve satisfying the 
given initial condition. 

12. Product of velocity times distance constant, equal to 2, 
y(0) = 2. 

13. Distance = Velocity X Time, y(l) = 1 

14. Square of the distance plus square of the velocity equal 
to 1 , initial distance 1 / V2 

15. Parachutist. Two forces act on a parachutist, the 
attraction by the earth mg (m = mass of person plus 
equipment, g = 9.8 m/sec 2 the acceleration of gravity) 
and the air resistance, assumed to be proportional to the 
square of the velocity v(t). Using Newton’s second law 
of motion (mass X acceleration = resultant of the forces), 
set up a model (an ODE for v(t )). Graph a direction field 
(choosing m and the constant of proportionality equal to 1). 
Assume that the parachute opens when v = 10 m/sec. 
Graph the corresponding solution in the field. What is the 
limiting velocity? Would the parachute still be sufficient 
if the air resistance were only proportional to u(r)? 


CHAP. 1 First-Order ODEs 

16. CAS PROJECT. Direction Fields. Discuss direction 
fields as follows. 

(a) Graph portions of the direction field of the ODE (2) 
(see Fig. 7), for instance, —5 £ x £ 2, —1 £ y £ 5. 
Explain what you have gained by this enlargement of 
the portion of the field. 

(b) Using implicit differentiation, find an ODE with 
the general solution x 2 + 9y 2 = c (y > 0). Graph its 
direction field. Does the field give the impression 
that the solution curves may be semi-ellipses? Can you 
do similar work for circles? Hyperbolas? Parabolas? 
Other curves? 

(c) Make a conjecture about the solutions of y ' = ~x/y 
from the direction field. 

(d) Graph the direction field of y = — \y and some 
solutions of your choice. How do they behave? Why 
do they decrease for y > 0? 



This is the simplest method to explain numerically solving 
an ODE, more precisely, an initial value problem (IVP). 
(More accurate methods based on the same principle are 
explained in Sec. 21.1.) Using the method, to get a feel for 
numerics as well as for the nature of IVPs, solve the IVP 
numerically with a PC or a calculator, 10 steps. Graph the 
computed values and the solution curve on the same 
coordinate axes. 



y = 

: y. 

y(0) = l. 






y = 

: y. 

v(0) = l. 






y = 

= O’ 






h = 0.1 


y = 

- x — tanh x 



y = 

= - 

5x 4 y 2 , y(0) 



h = 0.2 


y = 

= l/d + xf 

1.] Separable ODEs. Modeling 

Many practically useful ODEs can be reduced to the form 

(l) g(y)y' = f(x) 

by purely algebraic manipulations. Then we can integrate on both sides with respect to x, 

( 2 ) 


f(x) dx + c. 

On the left we can switch to y as the variable of integration. By calculus, y dx = dy, so that 

( 3 ) 


f(x ) dx + c. 

If / and g are continuous functions, the integrals in (3) exist, and by evaluating them we 
obtain a general solution of (1). This method of solving ODEs is called the method of 
separating variables, and (1) is called a separable equation, because in (3) the variables 
are now separated: x appears only on the right and y only on the left. 

E X A M Separable ODE 

The ODE y = 1 + y 2 is separable because it can be written 

= dx. By integration, arctan y = x + c or y = tan (x + c ). 

1 + y 2 

It is very important to introduce the constant of integration immediately when the integration is performed. 
If we wrote arctan y = x, then y = tan x, and then introduced c, we would have obtained y = tan x + c, which 
is not a solution (when c =£ 0). Verify this. 

SEC. 1.3 Separable ODEs. Modeling 





Separable ODE 

The ODE y = (x + 1 )e~ x y 2 is separable; we obtain y~ 2 dy = (x + \)e~ x dx. 
By integration, — y -1 = —Or + 2)e~ x + c, y = 


(x + 2)e — c 

Initial Value Problem (IVP). Bell-Shaped Curve 

Solve y' = —2xy,y(0) = 1.8. 

Solution. By separation and integration. 


— = —2 x dx, 

lny = — x 2 + c, y = ce . 

This is the general solution. From it and the initial condition, y(0) = ce u = c = 1.8. Hence the IVP has the 
solution y = 1.8e ~ x . This is a particular solution, representing a bell-shaped curve (Fig. 10). 

Fig. 10. Solution in Example 3 (bell-shaped curve) 


The importance of modeling was emphasized in Sec. 1.1, and separable equations yield 
various useful models. Let us discuss this in terms of some typical examples. 

Radiocarbon Dating 2 

In September 1991 the famous Iceman (Oetzi), a mummy from the Neolithic period of the Stone Age found in 
the ice of the Oetztal Alps (hence the name “Oetzi”) in Southern Tyrolia near the Austrian-Italian border, caused 
a scientific sensation. When did Oetzi approximately live and die if the ratio of carbon X gC to carbon 1 §C in 
this mummy is 52.5% of that of a living organism? 

Physical Information. In the atmosphere and in living organisms, the ratio of radioactive carbon gC (made 
radioactive by cosmic rays) to ordinary carbon *§0 is constant. When an organism dies, its absorption of X gC 
by breathing and eating terminates. Hence one can estimate the age of a fossil by comparing the radioactive 
carbon ratio in the fossil with that in the atmosphere. To do this, one needs to know the half-life of gC, which 
is 5715 years ( CRC Handbook of Chemistry and Physics, 83rd ed., Boca Raton: CRC Press, 2002, page 11-52, 
line 9). 

Solution. Modeling. Radioactive decay is governed by the ODE y' = ky (see Sec. 1.1, Example 5). By 
separation and integration (where t is time and yg is the initial ratio of gC to gC) 

— = kdt, In \y\ = kt + c, y = yoe kt (yo = <? c )- 


2 Method by WILLARD FRANK LIBBY (1908-1980), American chemist, who was awarded for this work 
the 1960 Nobel Prize in chemistry. 


CHAP. 1 First-Order ODEs 


Next we use the half-life H = 5715 to determine k. When t = H, half of the original substance is still present. Thus, 

J.TT In 0.5 0.693 

y 0 e kH = 0.5^o, e kH = 0.5, k = = — = -0.0001213. 

H 5715 

Finally, we use the ratio 52.5% for determining the time t when Oetzi died (actually, was killed), 

e kt = £ -0 - 0001213t = 0.525, t = = 5312. Answer: About 5300 years ago. 

-0.0001213 J 6 

Other methods show that radiocarbon dating values are usually too small. According to recent research, this is 
due to a variation in that carbon ratio because of industrial pollution and other factors, such as nuclear testing. 

Mixing Problem 

Mixing problems occur quite frequently in chemical industry. We explain here how to solve the basic model 
involving a single tank. The tank in Fig. 1 1 contains 1000 gal of water in which initially 100 lb of salt is dissolved. 
Brine runs in at a rate of 10 gal/min, and each gallon contains 5 lb of dissoved salt. The mixture in the tank is 
kept uniform by stirring. Brine runs out at 10 gal/min. Find the amount of salt in the tank at any time t. 

Solution. Step 1. Setting up a model. Let y(f) denote the amount of salt in the tank at time t. Its time rate 
of change is 

y — Salt inflow rate — Salt outflow rate Balance law. 

5 lb times 10 gal gives an inflow of 50 lb of salt. Now, the outflow is 10 gal of brine. This is 10/1000 = 0.01 
(= 1%) of the total brine content in the tank, hence 0.01 of the salt content y(t), that is, 0.01 y(t). Thus the 
model is the ODE 

(4) y = 50 - O.Oly = -0.01(y - 5000). 

Step 2. Solution of the model. The ODE (4) is separable. Separation, integration, and taking exponents on both 
sides gives 

— - = -0.01 dt. In Iv - 5000 1 = -O.Olf + c*, y- 5000 = ce~ omt . 

y — 5000 

Initially the tank contains 100 lb of salt. Hence y(0) = 100 is the initial condition that will give the unique 
solution. Substituting y = 100 and t = 0 in the last equation gives 100 — 5000 = ce = c. Hence c = —4900. 
Hence the amount of salt in the tank at time t is 

(5) y(t) = 5000 - 4900e~ o olt . 

This function shows an exponential approach to the limit 5000 lb; see Fig. 11. Can you explain physically that 
y(r) should increase with time? That its limit is 5000 lb? Can you see the limit directly from the ODE? 

The model discussed becomes more realistic in problems on pollutants in lakes (see Problem Set 1.5, Prob. 35) 
or drugs in organs. These types of problems are more difficult because the mixing may be imperfect and the flow 
rates (in and out) may be different and known only very roughly. 


Fig. 11. Mixing problem in Example 5 

SEC. 1.3 Separable ODEs. Modeling 



Heating an Office Building (Newton’s Law of Cooling 3 ) 

Suppose that in winter the daytime temperature in a certain office building is maintained at 70°F. The heating 
is shut off at 10 P.M. and turned on again at 6 a.m. On a certain day the temperature inside the building at 2 A.M. 
was found to be 65 °F. The outside temperature was 50°F at 10 P.M. and had dropped to 40°F by 6 a.m. What 
was the temperature inside the building when the heat was turned on at 6 A.M.? 

Physical information. Experiments show that the time rate of change of the temperature T of a body B (which 
conducts heat well, for example, as a copper ball does) is proportional to the difference between T and the 
temperature of the surrounding medium (Newton’s law of cooling). 

Solution. Step 1. Setting up a model. Let T{t) be the temperature inside the building and T A the outside 
temperature (assumed to be constant in Newton’s law). Then by Newton’s law, 

dT , 

(6) — = k(T ~ T a . ). 


Such experimental laws are derived under idealized assumptions that rarely hold exactly. However, even if a 
model seems to fit the reality only poorly (as in the present case), it may still give valuable qualitative information. 
To see how good a model is, the engineer will collect experimental data and compare them with calculations 
from the model. 

Step 2. General solution. We cannot solve (6) because we do not know T A , just that it varied between 50°F 
and 40°F, so we follow the Golden Rule: If you cannot solve your problem, try to solve a simpler one. We 
solve (6) with the unknown function T A replaced with the average of the two known values, or 45 °F. For physical 
reasons we may expect that this will give us a reasonable approximate value of T in the building at 6 a.m. 

For constant T A = 45 (or any other constant value) the ODE (6) is separable. Separation, integration, and 
taking exponents gives the general solution 

t f = k dt. In | T — 45 1 = kt + c*, T(t) = 45 + ce kt (c = e c '). 

Step 3. Particular solution. We choose 10 P.M. to be t = 0. Then the given initial condition is 7(0) = 70 and 
yields a particular solution, call it T p . By substitution, 

T( 0) = 45 + ce° = 70, c = 70 - 45 = 25, T v (t) = 45 + 25e fct . 

Step 4. Determination ofk. We use 7(4) = 65, where t = 4 is 2 a.m. Solving algebraically for k and inserting 
k into T V (J) gives (Fig. 12) 

T p (A) = 45 + 25e 4fc = 65, e 4fc = 0.8, k = \ In 0.8 = -0.056, T v {t) = 45 + 25e~ 0056t . 

Fig. 12. Particular solution (temperature) in Example 6 

3 Sir ISAAC NEWTON (1642-1727), great English physicist and mathematician, became a professor at 
Cambridge in 1669 and Master of the Mint in 1699. He and the German mathematician and philosopher 
GOTTFRIED WILHELM LEIBNIZ (1646-1716) invented (independently) the differential and integral calculus. 
Newton discovered many basic physical laws and created the method of investigating physical problems by 
means of calculus. His Philosophiae naturalis principia mathematica (. Mathematical Principles of Natural 
Philosophy, 1687) contains the development of classical mechanics. His work is of greatest importance to both 
mathematics and physics. 


CHAP. 1 First-Order ODEs 


Step 5. Answer and interpretation. 6 A.M. is r = 8 (namely. 8 hours after 10 P.M.), and 

T p ( 8) = 45 + 25e -0 ' 056 ' 8 = 61[°F], 

Hence the temperature in the building dropped 9°F, a result that looks reasonable. 

Leaking Tank. Outflow of Water Through a Hole (Torricelli’s Law) 

This is another prototype engineering problem that leads to an ODE. It concerns the outflow of water from a 
cylindrical tank with a hole at the bottom (Fig. 13). You are asked to find the height of the water in the tank at 
any time if the tank has diameter 2 m, the hole has diameter 1 cm, and the initial height of the water when the 
hole is opened is 2.25 m. When will the tank be empty? 

Physical information. Under the influence of gravity the outflowing water has velocity 

(7) v(t) = 0.600 \/2 gh(t) (Torricelli’s law 4 ), 

where h(t) is the height of the water above the hole at time t, and g = 980cm/sec 2 — 32.17 ft/sec 2 is the 
acceleration of gravity at the surface of the earth. 

Solution. Step 1. Setting up the model. To get an equation, we relate the decrease in water level h(t ) to the 
outflow. The volume AV of the outflow during a short time At is 

AV = Av At (A = Area of hole). 

AV must equal the change AV* of the volume of the water in the tank. Now 

AV* = — B Ah ( B = Cross-sectional area of tank) 

where Ah (> 0) is the decrease of the height h{t) of the water. The minus sign appears because the volume of 
the water in the tank decreases. Equating AV and AV* gives 

-B Ah = Av At. 

We now express v according to Torricelli’s law and then let At (the length of the time interval considered) 
approach 0 — this is a standard way of obtaining an ODE as a model. That is, we have 

Ah A 
~At = ~~B V 

-f 0.600V2 gh{t) 

and by letting At — > 0 we obtain the ODE 

— = -26.56 -VS. 
dt B 

where 26.56 = O.6OOV2 ■ 980. This is our model, a first-order ODE. 

Step 2. General solution. Our ODE is separable. A/B is constant. Separation and integration gives 

dh A A 

— — = —26.56 — dt and 2 \fh — c* — 26.56 — t. 

Vh B B 

Dividing by 2 and squaring gives h = (c — 1 3. 2SAt/B) 2 . Inserting 13.28 A/B = 13.28 • 0.5 2 7r/100 2 7r = 0.000332 
yields the general solution 

hit) = (c - 0.000 332tf. 

4 EVANGELISTA TORRICELLI (1608-1647), Italian physicist, pupil and successor of GALILEO GALILEI 
(1564-1642) at Florence. The “contraction factor” 0.600 was introduced by J. C. BORDA in 1766 because the 
stream has a smaller cross section than the area of the hole. 

SEC. 1.3 Separable ODEs. Modeling 


Step 3. Particular solution. The initial height (the initial condition) is h{ 0) = 225 cm. Substitution of t = 0 
and h = 225 gives from the general solution c 2, = 225, c = 15.00 and thus the particular solution (Fig. 13) 

hJt) = (15.00 - 0.0003320 

Step 4. Tank empty. h p (t ) = 0 if t = 15.00/0.000332 = 45,181 
Here you see distinctly the importance of the choice of units- 
in which time is measured in seconds! We used g = 980 cm/sec 2 

= 12.6 [hours]. 

we have been working with the cgs system, 

Step 5. Checking. Check the result. 

2.25 m 


at ti 

. i 







250 - 
200 - 
150 - 
100 - 
50 - 

0 I I I ! J 

0 10000 30000 50000 t 

Tank Water level h(t) in tank 

Fig. 13 Example 7. Outflow from a cylindrical tank (“leaking tank"). 
Torricelli's law 

Extended Method: Reduction to Separable Form 

Certain nonseparable ODEs can be made separable by transformations that introduce for 
y a new unknown function. We discuss this technique for a class of ODEs of practical 
importance, namely, for equations 

( 8 ) 

Here, /is any (differentiable) function of y/x, such as sin(y/x), (y/x) 4 , and so on. (Such 
an ODE is sometimes called a homogeneous ODE, a term we shall not use but reserve 
for a more important purpose in Sec. 1.5.) 

The form of such an ODE suggests that we set y/x = u\ thus, 

(9) y = ux and by product differentiation y' = u x + u. 

Substitution into y = f(y/x) then gives u x + u = f{u) or u x = f(u) — u. We see that 
if f{u) w f 0, this can be separated: 

du dx 

f(u) — u x 

( 10 ) 


CHAP. 1 First-Order ODEs 

EXAMPLE 8 Reduction to Separable Form 


/ 2 2 
2xyy = y — x . 

Solution. To get the usual explicit form, divide the given equation by 2 xy, 

2 2 

y -x y x 

2 xy 2x 2 y 

Now substitute y and y from (9) and then simplify by subtracting u on both sides. 

, u 1 

U X + u = — — — , 
2 2 u 

, u 1 —u — 1 
2 2 u 2u 

You see that in the last equation you can now separate the variables, 
2 u du 

1 + u 



By integration. 

In (1 + w 2 ) = -In \x\ + c* = In 

+ c*. 

Take exponents on both sides to get 1 + u 2 = c/x or 1 + (y/x) 2 = c/x. Multiply the last equation by x 2 to 
obtain (Fig. 14) 

2 , 2 

x + y = cx. 


- y = - 

This general solution represents a family of circles passing through the origin with centers on the x-axis. 

Fig. 14. General solution (family of circles) in Example 8 


1. CAUTION! Constant of integration. Why is it 

important to introduce the constant of integration 
immediately when you integrate? 



Find a general solution. Show the steps of derivation. Check 
your answer by substitution. 

2. y y + x =0 

3. y = sec y 

4. y sin 27 tx = Try cos 2ttx 

5. yy + 36x = 0 

6. y = e*-y 

7. xy' = y + 2x 3 sin 2 — (Set y/x = u) 

8. y = (y + 4x) 2 (Set y + 4x = v) 

9. xy = y z + y (Set y/x = u) 

10. xy' = x + y (Sety/.r = u) 


Solve the IVP. Show the steps of derivation, beginning with 
the general solution. 


11. xy' + y = 0, y(4) = 6 

12. y = 1 + 4y 2 , y(l) = 0 

13. y'cosh 2 * = sin 2 y, y(0) = \t t 

14. dr/dt = —2 tr, r( 0) = r 0 

15. y' = -4 x/y, y(2) = 3 

16. y' = (x + y - 2) 2 , y(0) = 2 

(Set v = x + y - 2) 

17. xy' = y + 3x 4 cos 2 (y/x), y(l) = 0 
(Set y/x = u ) 

18. Particular solution. Introduce limits of integration in 
(3) such that y obtained from (3) satisfies the initial 
condition y(x 0 ) = yo- 

SEC. 1.3 Separable ODEs. Modeling 




19. Exponential growth. If the growth rate of the number 
of bacteria at any time t is proportional to the number 
present at t and doubles in 1 week, how many bacteria 
can be expected after 2 weeks? After 4 weeks? 

20. Another population model. 

(a) If the birth rate and death rate of the number of 
bacteria are proportional to the number of bacteria 
present, what is the population as a function of time. 

(b) What is the limiting situation for increasing time? 
Interpret it. 

21. Radiocarbon dating. What should be the J |C content 
(in percent of y 0 ) of a fossilized tree that is claimed to 
be 3000 years old? (See Example 4.) 

22. Linear accelerators are used in physics for 
accelerating charged particles. Suppose that an alpha 
particle enters an accelerator and undergoes a constant 
acceleration that increases the speed of the particle 
from 10 3 m/sec to 10 4 m/sec in 10 -3 sec. Find the 
acceleration a and the distance traveled during that 
period of 10 -3 sec. 

23. Boyle-Mariotte’s law for ideal gases. 5 Experiments 
show for a gas at low pressure p (and constant 
temperature) the rate of change of the volume V(p) 
equals ~V/p. Solve the model. 

24. Mixing problem. A tank contains 400 gal of brine 
in which 100 lb of salt are dissolved. Fresh water runs 
into the tank at a rate of 2 gal/min.The mixture, kept 
practically uniform by stirring, runs out at the same 
rate. How much salt will there be in the tank at the 
end of 1 hour? 

25. Newton’s law of cooling. A thermometer, reading 
5°C, is brought into a room whose temperature is 22°C. 
One minute later the thermometer reading is 12°C. 
How long does it take until the reading is practically 
22°C, say, 21.9°C? 

26. Gompertz growth in tumors. The Gompertz model 
is y = —Ay In y (A > 0), where y(t) is the mass of 
tumor cells at time t. The model agrees well with 
clinical observations. The declining growth rate with 
increasing y > 1 corresponds to the fact that cells in 
the interior of a tumor may die because of insufficient 
oxygen and nutrients. Use the ODE to discuss the 
growth and decline of solutions (tumors) and to find 
constant solutions. Then solve the ODE. 

27. Dryer. If a wet sheet in a dryer loses its moisture at 
a rate proportional to its moisture content, and if it 
loses half of its moisture during the first 10 min of 

drying, when will it be practically dry, say, when will 
it have lost 99% of its moisture? First guess, then 

28. Estimation. Could you see, practically without calcu- 
lation, that the answer in Prob. 27 must lie between 
60 and 70 min? Explain. 

29. Alibi? Jack, arrested when leaving a bar, claims that 
he has been inside for at least half an hour (which 
would provide him with an alibi). The police check 
the water temperature of his car (parked near the 
entrance of the bar) at the instant of arrest and again 
30 min later, obtaining the values 190°F and 110°F, 
respectively. Do these results give Jack an alibi? 
(Solve by inspection.) 

30. Rocket. A rocket is shot straight up from the earth, 
with a net acceleration (= acceleration by the rocket 
engine minus gravitational pullback) of 7fm/sec 2 
during the initial stage of flight until the engine cut out 
at t — 10 sec. How high will it go, air resistance 

31. Solution curves of y' = g(y/x). Show that any 
(nonvertical) straight line through the origin of the 
xy-plane intersects all these curves of a given ODE at 
the same angle. 

32. Friction. If a body slides on a surface, it experiences 
friction F (a force against the direction of motion). 
Experiments show that |E| = /a|(V| (Coulomb’s 6 law of 
kinetic friction without lubrication), where N is the 
normal force (force that holds the two surfaces together; 
see Fig. 15) and the constant of proportionality p is 
called the coefficient of kinetic friction. In Fig. 15 
assume that the body weighs 45 nt (about 10 lb; see 
front cover for conversion), p = 0.20 (corresponding 
to steel on steel), a = 30°, the slide is 10 m long, the 
initial velocity is zero, and air resistance is 
negligible. Find the velocity of the body at the end 
of the slide. 

5 R0BERT BOYLE (1627-1691), English physicist and chemist, one of the founders of the Royal Society. EDME MARIOTTE (about 
1620-1684), French physicist and prior of a monastry near Dijon. They found the law experimentally in 1662 and 1676, respectively. 

e CHARLES AUGUSTIN DE COULOMB (1736-1806). French physicist and engineer. 


CHAP. 1 First-Order ODEs 

33. Rope. To tie a boat in a harbor, how many times 
must a rope be wound around a bollard (a vertical 
rough cylindrical post fixed on the ground) so that a 
man holding one end of the rope can resist a force 
exerted by the boat 1000 times greater than the man 
can exert? First guess. Experiments show that the 
change AS of the force 5 in a small portion of the 
rope is proportional to S and to the small angle \(f> 
in Fig. 16. Take the proportionality constant 0.15. 
The result should surprise you! 

34. TEAM PROJECT. Family of Curves. A family of 
curves can often be characterized as the general 
solution of y = f(x, y). 

(a) Show that for the circles with center at the origin 
we get y' = —x/y. 

(b) Graph some of the hyperbolas xy = c. Find an 
ODE for them. 

(c) Find an ODE for the straight lines through the 

(d) You will see that the product of the right sides of 
the ODEs in (a) and (c) equals — 1 . Do you recognize 

this as the condition for the two families to be 
orthogonal (i.e., to intersect at right angles)? Do your 
graphs confirm this? 

(e) Sketch families of curves of your own choice and 
find their ODEs. Can every family of curves be given 
by an ODE? 

35. CAS PROJECT. Graphing Solutions. A CAS can 

usually graph solutions, even if they are integrals that 
cannot be evaluated by the usual analytical methods of 

(a) Show this for the five initial value problems 
y — e~ x , y(0) = 0, ±1, ±2 graphing all five curves 
on the same axes. 

(b) Graph approximate solution curves, using the first 
few terms of the Maclaurin series (obtained by term- 
wise integration of that of y') and compare with the 
exact curves. 

(c) Repeat the work in (a) for another ODE and initial 
conditions of your own choice, leading to an integral 
that cannot be evaluated as indicated. 

36. TEAM PROJECT. Torricelli’s Law. Suppose that 
the tank in Example 7 is hemispherical, of radius R, 
initially full of water, and has an outlet of 5 cm 2 cross- 
sectional area at the bottom. (Make a sketch.) Set 
up the model for outflow. Indicate what portion of 
your work in Example 7 you can use (so that it can 
become part of the general method independent of the 
shape of the tank). Find the time t to empty the tank 
(a) for any R, (b) for R = 1 m. Plot t as function of 
R. Find the time when h = R/2 (a) for any R, (b) for 
R = 1 m. 

Exact ODEs. Integrating Factors 

We recall from calculus that if a function u(x, y ) has continuous partial derivatives, its 
differential (also called its total differential') is 

du dll 

du = — dx H dy. 

dx dy 

From this it follows that if u(x, y) = c = const, then du = 0. 

For example, if u = x + x 2 y 3 = c, then 

du = (1 + 2xy 3 ) dx + 3x 2 y 2 dy = 0 


i dy 1 + 2 xy 3 
y dx 3 x 2 y 2 

SEC. 1.4 Exact ODEs. Integrating Factors 


an ODE that we can solve by going backward. This idea leads to a powerful solution 
method as follows. 

A first-order ODE Mix, y ) + Nix, y)y = 0, written as (use dy = y dx as in Sec. 1.3) 

(1) M(x, y) dx + N(x, y) dy = 0 

is called an exact differential equation if the differential form Mix, y) dx + N(x, y) dy 
is exact, that is, this form is the differential 

du du 

(2) du = — dx H dy 

dx dy 

of some function u(x, y). Then (1) can be written 

du = 0. 

By integration we immediately obtain the general solution of (1) in the form 
(3) u(x, y) = c. 

This is called an implicit solution, in contrast to a solution y = h(x) as defined in Sec. 
1.1, which is also called an explicit solution, for distinction. Sometimes an implicit solution 
can be converted to explicit form. (Do this for x z + y 2 = 1.) If this is not possible, your 
CAS may graph a figure of the contour lines (3) of the function m(x, y) and help you in 
understanding the solution. 

Comparing (1) and (2), we see that (1) is an exact differential equation if there is some 
function u(x, y) such that 

du du 

(4) (a) — = M, (b) — = N. 

dx dy 

From this we can derive a formula for checking whether (1) is exact or not, as follows. 

Let M and N be continuous and have continuous first partial derivatives in a region in 
the xy- plane whose boundary is a closed curve without self-intersections. Then by partial 
differentiation of (4) (see App. 3.2 for notation), 

dM _ d 2 u 
dy dy dx’ 

dN d 2 u 

dx dx dy 

By the assumption of continuity the two second partial derivaties are equal. Thus 

dM _ dN 
dy dx 

( 5 ) 


CHAP. 1 First-Order ODEs 


This condition is not only necessary but also sufficient for (1) to be an exact differential 
equation. (We shall prove this in Sec. 10.2 in another context. Some calculus books, for 
instance, [GenRef 12], also contain a proof.) 

If (1) is exact, the function u(x,y) can be found by inspection or in the following 
systematic way. From (4a) we have by integration with respect to x 

( 6 ) 


M dx + k(y ); 

in this integration, y is to be regarded as a constant, and k(y) plays the role of a “constant” 
of integration. To determine k(y), we derive du/dy from (6), use (4b) to get dk/dy, and 
integrate dk/dy to get k. (See Example 1, below.) 

Formula (6) was obtained from (4a). Instead of (4a) we may equally well use (4b). 
Then, instead of (6), we first have by integration with respect to y 



N dy + l(x). 

To determine l(x), we derive du/dx from (6*), use (4a) to get dl/dx, and integrate. We 
illustrate all this by the following typical examples. 

An Exact ODE 


(7) cos ( x + y) dx + (3y 2 + 2 y + cos (x + y)) dy = 0. 

Solution. Step 1. Test for exactness. Our equation is of the form (1) with 

M = cos (x + y), 

N = 3 y 2 + 2y + cos ( x + y). 


— = “sin (.v + y), 


— = -sm (x + y). 

From this and (5) we see that (7) is exact. 

Step 2. Implicit general solution. From (6) we obtain by integration 

(8) u = I M dx + k(y) = | cos (x + y) dx + k(y ) = sin (x + y) + k(y). 

To find k(y), we differentiate this formula with respect to y and use formula (4b), obtaining 

du dk o 

— = cos (x + y) H = N = 3 y + 2y + cos (x + y). 

dy dy 

Hence dk/dy = 3y 2 + 2y. By integration, k = y 3 + v 2 + c*. Inserting this result into (8) and observing (3), 
we obtain the answer 

u(x, y) = sin (x + y) + y 3 + y 2 = c. 

SEC. 1.4 Exact ODEs. Integrating Factors 




Step 3. Checking an implicit solution. We can check by differentiating the implicit solution u(x, y) = c 
implicitly and see whether this leads to the given ODE (7): 

du du 9 

(9) du = — dx H dy = cos ( x + y) dx + (cos (x + y) + 3 y + 2y) dy = 0. 

dx By 

This completes the check. 

An Initial Value Problem 

Solve the initial value problem 

(10) (cosy sinhx + l) dx — sin y cosh x dy = 0, y(l) = 2. 

Solution. You may verify that the given ODE is exact. We find u. For a change, let us use (6*), 

u = — J sin y cosh x dy + /(x) = cos y cosh x + /(x). 

From this, dw/dx = cosy sinhx + dl/dx = M = cosy sinhx + 1 . Hence dl/dx = 1 . By integration, /(x) = x + c*. 
This gives the general solution w(x, y) = cos y cosh x + x = c. From the initial condition, cos 2 cosh 1 + 1 = 
0.358 = c. Hence the answer is cos y cosh x + x = 0.358. Figure 17 shows the particular solutions fore = 0, 0.358 
(thicker curve), 1, 2, 3. Check that the answer satisfies the ODE. (Proceed as in Example 1.) Also check that the 
initial condition is satisfied. 

Fig. 17. Particular solutions in Example 2 

WARNING! Breakdown in the Case of Nonexactness 

The equation — y dx + x dy = 0 is not exact because M = — y and N = x, so that in (5), BM/By = — 1 but 
BN/Bx = 1. Let us show that in such a case the present method does not work. From (6), 

f Bu dk 

u — \M dx + k(y) = —xy + k(y), hence — = -x H . 

J dy dy 

Now, Bu/By should equal N = x, by (4b). However, this is impossible because k(y) can depend only on y. Try 
(6*); it will also fail. Solve the equation by another method that we have discussed. 

Reduction to Exact Form. Integrating Factors 

The ODE in Example 3 is —y dx + x dy = 0. It is not exact. However, if we multiply it 
by 1/x 2 , we get an exact equation [check exactness by (5)!], 

( 11 ) 

— y dx + x dy 



— L dx H — dy = d ) = 0. 
X X \xj 

Integration of (11) then gives the general solution y/x = c = const. 


CHAP. 1 First-Order ODEs 


This example gives the idea. All we did was to multiply a given nonexact equation, say, 
(12) P(x, y) dx + Q(x, y) dy = 0, 

by a function F that, in general, will be a function of both x and y. The result was an equation 

(13) FPdx + FQdy = 0 

that is exact, so we can solve it as just discussed. Such a function F(x, y) is then called 

an integrating factor of (12). 

Integrating Factor 

The integrating factor in (1 1) is F = 1/jt 2 . Hence in this case the exact equation (13) is 

— y dx + xdy (y\ y 

FP dx + FQ dy = — : = d — ) = 0. Solution - = c. 

x z W * 

These are straight lines y = cx through the origin. (Note that x = 0 is also a solution of — y dx + xdy = 0.) 

It is remarkable that we can readily find other integrating factors for the equation — y dx + xdy = 0, namely, 
1/y 2 , 1 /fry), and I /(x 2 + y 2 ), because 


~y dx + xdy 

— y dx + x dy 

~y dx + xdy 

= d arctan : 

How to Find Integrating Factors 

In simpler cases we may find integrating factors by inspection or perhaps after some trials, 
keeping (14) in mind. In the general case, the idea is the following. 

For M dx + N dy = 0 the exactness condition (5) is dM/dy = dN/dx. Hence for (13), 
FP dx + FQ dy = 0, the exactness condition is 


d d 

— {FP) = —{FQ). 
dy ax 

By the product rule, with subscripts denoting partial derivatives, this gives 

F y P + FP y - F X Q + FQ X . 

In the general case, this would be complicated and useless. So we follow the Golden Rule: 
If you cannot solve your problem, try to solve a simpler one — the result may be useful 
(and may also help you later on). Hence we look for an integrating factor depending only 
on one variable: fortunately, in many practical cases, there are such factors, as we shall 
see. Thus, let F = Fix). Then F y = 0, and F x = F' = dF/dx, so that (15) becomes 

FP y = F'Q + FQ X . 

Dividing by FQ and reshuffling terms, we have 

F dx 

= R, 

Q\dy dxj 



SEC. 1.4 Exact ODEs. Integrating Factors 





This proves the following theorem. 

Integrating Factor F(x) 

If ( 12) is such that the right side R of ( 16) depends only on x, then (12) has an 
integrating factor F = F(x), which is obtained by integrating (16) and taking 
exponents on both sides. 


Fix) = exp R(x) dx. 

Similarly, if F* = F*(y), then instead of (16) we get 


1 dF* 

= R*, 

F* dy 


and we have the companion 

Integrating Factor F*(y) 

If (12) is such that the right side R* of ( 18) depends only on y, then (12) has an 
integrating factor F* = F*(y), which is obtained from (18) in the form 

( 19 ) 

F*(y) = exp 

R*(y) dy. 

Application of Theorems 1 and 2. Initial Value Problem 

Using Theorem 1 or 2, find an integrating factor and solve the initial value problem 

(20) (e x+y + ye y ) dx + (xe y - l) dy - 0, y(0) = -1 

Solution. Step 1. Nonexactness. The exactness check fails: 

3 P 3 x+y v v SQ s v „ 

— = — (e y + ye v ) = e y + e v + ye v but — = — ( xe y - 1) = e y . 
dy dy dx dx 

Step 2. Integrating factor. General solution. Theorem 1 fails because R [the right side of (16)] depends on 
both x and y. 


_ jf3P _ _ 

Q \ dy dx 
Try Theorem 2. The right side of (18) is 

(e x 

e y + ye v - i 


i_t dQ _ dP\ 
P\dx dy J 


- (e y - e x 

e y - ye v ) = -1. 

Hence (19) gives the integrating factor F*(y) = e y . From this result and (20) you get the exact equation 

( e x + y) dx + (x — e ~ y ) dy = 0. 


CHAP. 1 First-Order ODEs 

Test for exactness; you will get 1 on both sides of the exactness condition. By integration, using (4a), 

u = | {e x + y) dx = e x + xy + k(y). 

Differentiate this with respect to y and use (4b) to get 


du dk 

— = x -\ = N = x — e y , 

dy dy 


k — e y + c*. 

Hence the general solution is 

u{x, y) — e x + xy + e y = c. 

Setp 3 . Particular solution. The initial condition y(0) = — 1 gives u{ 0, — 1) = 1 + 0 + e = 3.72. Hence the 
answer is e x + xy + e~ y = 1 + e = 3.72. Figure 18 shows several particular solutions obtained as level curves 
of u(x, y) = c, obtained by a CAS, a convenient way in cases in which it is impossible or difficult to cast a 
solution into explicit form. Note the curve that (nearly) satisfies the initial condition. 

Step 4. Checking. Check by substitution that the answer satisfies the given equation as well as the initial 



Test for exactness. If exact, solve. If not, use an integrating 
factor as given or obtained by inspection or by the theorems 
in the text. Also, if an initial condition is given, find the 
corresponding particular solution. 

1. 2xy dx + x 2 dy = 0 

2. x 3 dx + y 3 dy = 0 

3. sin x cos y dx + cos x sin y dy = 0 

4 . e 3e (dr + 3rd6) = 0 

5. (x z + y 2 )dx — 2xy dy = 0 

6. 3(y + l) dx = 2xdy, (y + l).r -4 

7. 2x tan y dx + sec 2 y dy = 0 

8. e x (cos y dx — sin y dy) = 0 

9. e 2x (2cosydx — sinyrfy) = 0, _v(0) = 0 

10. y dx + [y + tan {x + y)] dy = 0, cos ( x + y) 

11. 2 cosh x cos y dx — sinh x sin y dy 

12. (2xy dx + dy)e x = 0, y(0) = 2 

13. e~ v dx + e- x (-e~ v + \)dy = 0, F = e x+v 

14. ( a + l)y dx + {b + \)xdy — 0, y(l) = 1, 

F = x a y b 

15. Exactness. Under what conditions for the constants a, 
b, k, l is (ax + by) dx + ( kx + ly) dy = 0 exact? Solve 
the exact ODE. 

SEC. 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 


16. TEAM PROJECT. Solution by Several Methods. 

Show this as indicated. Compare the amount of work. 

(a) e v (sm\\ x dx + cosh xdy) = Oas an exact ODE 
and by separation. 

(b) (1 + 2x) cos ydx + dy/cosy = Oby Theorem 2 
and by separation. 

(c) (x 2 + y 2 ) dx — 2xy dy = 0 by Theorem 1 or 2 and 
by separation with v = y/x. 

(d) 3jc 2 y dx + 4x 3 dy = 0 by Theorems 1 and 2 and 
by separation. 

(e) Search the text and the problems for further ODEs 
that can be solved by more than one of the methods 
discussed so far. Make a list of these ODEs. Find 
further cases of your own. 

17. WRITING PROJECT. Working Backward. 

Working backward from the solution to the problem 
is useful in many areas. Euler, Lagrange, and other 
great masters did it. To get additional insight into 
the idea of integrating factors, start from a u(x, y) of 
your choice, find du = 0, destroy exactness by 
division by some F(x, y), and see what ODE’s 
solvable by integrating factors you can get. Can you 
proceed systematically, beginning with the simplest 
F(x, y)? 

18. CAS PROJECT. Graphing Particular Solutions. 

Graph particular solutions of the following ODE, 
proceeding as explained. 

(21) dy — y 2 sin.r dx = 0. 

(a) Show that (21) is not exact. Find an integrating 
factor using either Theorem 1 or 2. Solve (21). 

(b) Solve (21) by separating variables. Is this simpler 
than (a)? 

(c) Graph the seven particular solutions satisfying the 
following initial conditions y(0) = 1, y(7r/2) = ±|, 
±|, ± 1 (see figure below). 

(d) Which solution of (21) do we not get in (a) or (b)? 

Particular solutions in CAS Project 18 

Linear ODEs. Bernoulli Equation. 

Population Dynamics 

Linear ODEs or ODEs that can be transformed to linear form are models of various 
phenomena, for instance, in physics, biology, population dynamics, and ecology, as we 
shall see. A first-order ODE is said to be linear if it can be brought into the form 

(1) y + p(x)y = r(x), 

by algebra, and nonlinear if it cannot be brought into this form. 

The defining feature of the linear ODE (1) is that it is linear in both the unknown 
function y and its derivative y = dy/dx , whereas p and r may be any given functions of 
x. If in an application the independent variable is time, we write t instead of x. 

If the first term is f(x)y (instead of y ), divide the equation by fix) to get the standard 
form (1), with y as the first term, which is practical. 

For instance, y cos x + y sin x = x is a linear ODE, and its standard form is 
y + y tan x = x sec x. 

The function r(x) on the right may be a force, and the solution y(x) a displacement in 
a motion or an electrical current or some other physical quantity. In engineering, r(x) is 
frequently called the input, and y(x) is called the output or the response to the input (and, 
if given, to the initial condition). 


CHAP. 1 First-Order ODEs 

Homogeneous Linear ODE. We want to solve (1) in some interval a < x < b, call 
it J, and we begin with the simpler special case that r(x) is zero for all x in J. (This is 
sometimes written r(x) = 0.) Then the ODE (1) becomes 

( 2 ) 

y + P(x)y = 0 

and is called homogeneous. By separating variables and integrating we then obtain 


= —p(x)dx. 


In \y\ = - 

p(x)dx + c*. 

Taking exponents on both sides, we obtain the general solution of the homogeneous 
ODE (2), 

( 3 ) 

y(x) = ce~^ x)dx 

(c = ±e c when y - ~ 0); 

here we may also choose c = 0 and obtain the trivial solution y(x) = 0 for all x in that 

Nonhomogeneous Linear ODE. We now solve (1) in the case that r(x) in (1) is not 
everywhere zero in the interval / considered. Then the ODE (1) is called nonhomogeneous. 
It turns out that in this case, (1) has a pleasant property; namely, it has an integrating factor 
depending only on x. We can find this factor F(x) by Theorem 1 in the previous section 
or we can proceed directly, as follows. We multiply (1) by F(x), obtaining 

(1*) Fy' + pFy = rF. 

The left side is the derivative (Fy)' = F'y + Fy' of the product Fy if 

pFy = F'y, thus pF = F' . 

By separating variables, dF/F = p dx. By integration, writing h = J p dx, 

P dx, 


F = e h . 

In |f| = h = 

With this F and h' = p, Eq. (1*) becomes 

h f i j f h h f i / h\f / h h 

e y + h e y = e y + (e ) y = (e y) = re . 

By integration, 

e y 

e r dx + c. 

Dividing by e , we obtain the desired solution formula 

( 4 ) 

y(x) = e 


e r dx + c ), h = 

p(x) dx. 

SEC. 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 




This reduces solving (1) to the generally simpler task of evaluating integrals. For ODEs 
for which this is still difficult, you may have to use a numeric method for integrals from 
Sec. 19.5 or for the ODE itself from Sec. 21.1. We mention that h has nothing to do with 
h{x) in Sec. 1.1 and that the constant of integration in h does not matter; see Prob. 2. 

The structure of (4) is interesting. The only quantity depending on a given initial 
condition is c. Accordingly, writing (4) as a sum of two terms, 


we see the following: 

y(x ) = e 


e h rdx + ce h . 

(5) Total Output = Response to the Input r + Response to the Initial Data. 

First-Order ODE, General Solution, Initial Value Problem 

Solve the initial value problem 

y + y tan x = sin 2x, y(0) = 1 . 

Solution. Here p = tan x, r = sin 2x = 2 sin cos x, and 

h = J p dx = | tan x dx = In | sec x\ . 

From this we see that in (4), 

e h = sec x, e~ h = cos x, e h r = (sec x){2 sin x cos x) = 2 sin x, 
and the general solution of our equation is 

y(jr) = cos x I 2 sin x dx + c ] = c cos x — 2 cos x. 

From this and the initial condition, 1 = c • 1 — 2 • l 2 ; thus c = 3 and the solution of our initial value problem 
is y = 3 cos x — 2 cos 2 x. Here 3 cos x is the response to the initial data, and —2 cos 2 jc is the response to the 
input sin 2jc. 

Electric Circuit 

Model the RL- circuit in Fig. 19 and solve the resulting ODE for the current I(t) A (amperes), where t is 
time. Assume that the circuit contains as an EMF E{t) (electromotive force) a battery of E = 48 V (volts), which 
is constant, a resistor of R = 1 1 D (ohms), and an inductor of L = 0.1 H (henrys), and that the current is initially 

Physical Laws. A current I in the circuit causes a voltage drop RI across the resistor (Ohm’s law) and 
a voltage drop Li' = L dl/dt across the conductor, and the sum of these two voltage drops equals the EMF 

(Kirchhoff’s Voltage Law, KVL). 

Remark. In general, KVL states that “The voltage (the electromotive force EMF) impressed on a closed 
loop is equal to the sum of the voltage drops across all the other elements of the loop.” For Kirchoff’s Current 
Law (KCL) and historical information, see footnote 7 in Sec. 2.9. 

Solution. According to these laws the model of the /?L-circuit is Li' + RI = E(t), in standard form 

R J = m 
L L ' 



CHAP. 1 First-Order ODEs 


We can solve this linear ODE by (4) with x = t, y = 7, p = R/ L, h = ( R/L)t , obtaining the general solution 

/ = + c \ 

By integration, 

( 7 ) 

/ r „(R/i)' , ,, 

/ = e~ (R ' Lyt - — - + c = - + ce-W*. 

\L R/L 


In our case, R/L = 11/0.1 = 110 and E(t) = 48/0.1 = 480 = const; thus, 

In modeling, one often gets better insight into the nature of a solution (and smaller roundoff errors) by inserting 
given numeric data only near the end. Here, the general solution (7) shows that the current approaches the limit 
E/R = 48/11 faster the larger R/L is, in our case, R/L = 11/0.1 = 110, and the approach is very fast, from 
below if 1(0) < 48/ 1 1 or from above if 1(0) > 48/ 1 1 . If 7(0) = 48/ 1 1 , the solution is constant (48/1 1 A). See 
Fig. 19. 

The initial value 7(0) = 0 gives 7(0) = E/R + c = 0, c = —E/R and the particular solution 
(8) 7 = -(1 - e _(R/Mt ), thus 7 = jj( 1 - e -110t ). 

R = 11 (1 

0.01 0.02 0.03 0.04 



Current I{t ) 


Fig. 19. RL-circuit 

Hormone Level 

Assume that the level of a certain hormone in the blood of a patient varies with time. Suppose that the time rate 
of change is the difference between a sinusoidal input of a 24-hour period from the thyroid gland and a continuous 
removal rate proportional to the level present. Set up a model for the hormone level in the blood and find its 
general solution. Find the particular solution satisfying a suitable initial condition. 

Solution. Step 1. Setting up a model. Let y{t) be the hormone level at time t. Then the removal rate is Ky(t). 
The input rate is A + B cos cot, where a) = 2tt/24 = 77 / 12 and A is the average input rate; here A B to make 
the input rate nonnegative. The constants A, B, K can be determined from measurements. Hence the model is the 
linear ODE 

y it) = In — Out = A + B cos cot — Ky(t), thus y' + Ky = A + B cos cot. 

The initial condition for a particular solution y part is y pa rt(0) = y 0 with t = 0 suitably chosen, for example, 
6:00 A.M. 

Step 2. General solution. In (4) we have p = K = const, h — Kt, and r = A + B cos cot. Hence (4) gives the 
general solution (evaluate f e Kt cos cot dt by integration by parts) 

SEC. 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 


y(t) = e 

A + B cos cot )dt + ce 



K cos ojt + oj sin cot 

+ ce 



K 2 + (77/ nf 

TTt TT . 777 

K cos 1 sin — 

12 12 12 

The last term decreases to 0 as t increases, practically after a short time and regardless of c (that is, of the initial 
condition). The other part of y(t) is called the steady-state solution because it consists of constant and periodic 
terms. The entire solution is called the transient-state solution because it models the transition from rest to the 
steady state. These terms are used quite generally for physical and other systems whose behavior depends on time. 

Step 3. Particular solution. Setting t = 0 in y(t) and choosing y 0 = 0, we have 

y(0) = 


K + c = 0, 


K K 2 + (tt/12) 2 77 
Inserting this result into y(t), we obtain the particular solution 

_ _A _ 


K K 2 + (tt/12) 2 

ypart^O "L 


K K 2 + (tt/12) 2 

TTt 7T 


K cos — H — — sin — — 1 — I 1- 

12 12 



K K 2 + (tt/12) : 

with the steady-state part as before. To plot _y paI1 we must specify values for the constants, say, A = B = 1 
and K = 0.05. Figure 20 shows this solution. Notice that the transition period is relatively short (although 
K is small), and the curve soon looks sinusoidal; this is the response to the input A + Bcos(jjTTf) = 

1 + COS (pj TTt). 






5 ■ 



Fig. 20. Particular solution in Example 3 

Reduction to Linear Form. Bernoulli Equation 

Numerous applications can be modeled by ODEs that are nonlinear but can be transformed 
to linear ODEs. One of the most useful ones of these is the Bernoulli equation 7 

( 9 ) 

y + P(x)y = g(x)y a 

(, a any real number). 

7 JAKOB BERNOULLI (1654—1705), Swiss mathematician, professor at Basel, also known for his contribution 
to elasticity theory and mathematical probability. The method for solving Bernoulli’s equation was discovered by 
Leibniz in 1696. Jakob Bernoulli’s students included his nephew NIKLAUS BERNOULLI (1687-1759), who 
contributed to probability theory and infinite series, and his youngest brother JOHANN BERNOULLI (1667-1748), 
who had profound influence on the development of calculus, became Jakob’s successor at Basel, and had among 
his students GABRIEL CRAMER (see Sec. 7.7) and LEONHARD EULER (see Sec. 2.5). His son DANIEL 
BERNOULLI (1700-1782) is known for his basic work in fluid flow and the kinetic theory of gases. 


CHAP. 1 First-Order ODEs 


If a = 0 or a = 1, Equation (9) is linear. Otherwise it is nonlinear. Then we set 

u(x) = [j(.r)] 1- “. 

We differentiate this and substitute y from (9), obtaining 

u = (1 - a)y~ a y' = (1 - a)y~ a (gy a - py). 

Simplification gives 

«'=(!- a)(g ~ 

where v 1_a = u on the right, so that we get the linear ODE 
(10) u + (1 — a)pu = (1 — a)g. 

For further ODEs reducible to linear form, see lnce’s classic [All] listed in App. 1. See 
also Team Project 30 in Problem Set 1.5. 

Logistic Equation 

Solve the following Bernoulli equation, known as the logistic equation (or Verhulst equation 8 ): 

(11) y = Ay - By 2 

Solution. Write (11) in the form (9), that is, 

y' - Ay = ~ By 2 

to see that a = 2, so that u = )> 1-a = y -1 . Differentiate this u and substitute y' from (11), 
u = ~y~ 2 y' = -y~\Ay - By 2 ) = B - Ay' 1 . 

The last term is — Ay -1 = —An. Hence we have obtained the linear ODE 

u + Au = B. 

The general solution is [by (4)] 

u = ce~ At + B/A. 

Since u = 1 /y, this gives the general solution of (11), 

( 12 ) 


Directly from (11) we see that y = 0 (y(t) = 0 for all t) is also a solution. 

(Fig. 21) 

8 PIERRE-FRAN£OIS VERHULST, Belgian statistician, who introduced Eq. (8) as a model for human 
population growth in 1838. 

SEC. 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 



Fig. 21. Logistic population model. Curves (9) in Example 4 with A/B = 4 

Population Dynamics 

The logistic equation (11) plays an important role in population dynamics, a field 
that models the evolution of populations of plants, animals, or humans over time t. 
If B = 0, then (11) is y' = dy/dt = Ay. In this case its solution (12) is y = (1 /c)e At 
and gives exponential growth, as for a small population in a large country (the 
United States in early times!). This is called Malthus’s law. (See also Example 3 in 
Sec. 1.1.) 

The term —By 2 in (11) is a “braking term” that prevents the population from growing 
without bound. Indeed, if we write y = Ay [ 1 — (B/A)y], we see that if y < A/B. then 
y > 0, so that an initially small population keeps growing as long as y < A/B. But if 
y > A/B, then y < 0 and the population is decreasing as long as y > A/B. The limit 
is the same in both cases, namely, A/B. See Fig. 21. 

We see that in the logistic equation (11) the independent variable t does not occur 
explicitly. An ODE y = fit, y) in which t does not occur explicitly is of the form 

(13) y' = fiy) 

and is called an autonomous ODE. Thus the logistic equation (11) is autonomous. 

Equation (13) has constant solutions, called equilibrium solutions or equilibrium 
points. These are determined by the zeros of fiy), because fiy) = 0 gives y = 0 by 
(13); hence y = const. These zeros are known as critical points of (13). An 
equilibrium solution is called stable if solutions close to it for some t remain close 
to it for all further t. It is called unstable if solutions initially close to it do not remain 
close to it as t increases. For instance, y = 0 in Fig. 21 is an unstable equilibrium 
solution, and y = 4 is a stable one. Note that (11) has the critical points y = 0 and 
y = A/B. 

Stable and Unstable Equilibrium Solutions. “Phase Line Plot” 

The ODE y' = (y — l)(y — 2) has the stable equilibrium solution yx = 1 and the unstable y 2 — 2, as the direction 
field in Fig. 22 suggests. The values y x and y 2 are the zeros of the parabola /(y) = (y — l)(y — 2) in the figure. 
Now, since the ODE is autonomous, we can “condense” the direction field to a “phase line plot” giving yx and 
y 2 , and the direction (upward or downward) of the arrows in the field, and thus giving information about the 
stability or instability of the equilibrium solutions. 


CHAP. 1 First-Order ODEs 

yU ) 

1 1 1 1 1 1 / ts.a 

t t t t t tt t t t 


tt 1 1 1 1 1 1 1 1 
/ / / / / / / / / / 


'//// ’ 0 : 5 ' 



4 / / / // / / / / / / 4 / / / / 







■ yi 

2.0 - 

1.5 - 


1.0 - 

0.5 - 


u 2 1 | 3* 2 

0.5 1.0 1.5 2.0 2.5 3.0 * 

(a) (6) (c) 

Fig. 22. Example 5. (A) Direction field. (B) “Phase line”. (C) Parabola f(y) 

A few further population models will be discussed in the problem set. For some more 
details of population dynamics, see C. W. Clark. Mathematical Bioeconomics : The 
Mathematics of Conservation 3rd ed. Hoboken, NJ, Wiley, 2010. 

Further applications of linear ODEs follow in the next section. 

-In x 

= l/x(not —x) and 

1. CAUTION! Show that e 

e — ln(secx) = cosx 

2. Integration constant. Give a reason why in (4) you may 
choose the constant of integration in fp dx to be zero. 




Find the general solution. If an initial condition is given, 
find also the corresponding particular solution and graph or 
sketch it. (Show the details of your work.) 

3. y' — y = 5.2 

4. y = 2y — 4x 

5. y' + ky = e~ kx 

6. y + 2y = 4 cos 2x, y{\l r) = 3 

7. xy = 2v + x 3 e x 

8. y' + ytanx = e“ 001l cosr, y(0) = 0 

9. y + ysinx = e cosx , y(0) = -2.5 

10. y' cos x + (3y — l)sec.r = 0, yi^Tr) — 4/3 

11 . y = (y - 2) cot x 

12. xy + 4y = 8x 4 , y( I ) = 2 

13. y = 6 (y — 2.5)tanh 1.5x 

14. CAS EXPERIMENT, (a) Solve the ODE y - y/x = 
—x _1 cos (l/x).Find an initial condition for which the 
arbitrary constant becomes zero. Graph the resulting 
particular solution, experimenting to obtain a good 
figure near x = 0. 

(b) Generalizing (a) from n = 1 to arbitrary n, solve the 
ODE y — ny/x = — x” -2 cos (1/x). Find an initial 
condition as in (a) and experiment with the graph. 



These properties are of practical and theoretical importance 
because they enable us to obtain new solutions from given 
ones. Thus in modeling, whenever possible, we prefer linear 
ODEs over nonlinear ones, which have no similar properties. 

Show that nonhomogeneous linear ODEs (1) and homo- 
geneous linear ODEs (2) have the following properties. 
Illustrate each property by a calculation for two or three 
equations of your choice. Give proofs. 

15. The sum + y 2 of two solutions yq and y 2 of the 
homogeneous equation (2) is a solution of (2), and so is 
a scalar multiple ay 1 for any constant a. These properties 
are not true for (1)! 

SEC. 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics 


16. y = 0 (that is, y(x) = 0 for all x, also written y(x) = 0) 
is a solution of (2) [not of (1) if r(x) A 0!], called the 

trivial solution. 

17. The sum of a solution of (1) and a solution of (2) is a 
solution of (1). 

18. The difference of two solutions of (1) is a solution of (2). 

19. If yi is a solution of (1), what can you say about cyi? 

20. If yi and y 2 are solutions of y[ + py i = tq and 
y 2 + py 2 — r 2 , respectively (with the same p\), what 
can you say about the sum yi + v 2 ? 

21. Variation of parameter. Another method of obtaining 
(4) results from the following idea. Write (3) as cy*, 
where y* is the exponential function, which is a solution 
of the homogeneous linear ODE y *' + py* = 0. 
Replace the arbitrary constant c in (3) with a function 
u to be determined so that the resulting function y — uy* 
is a solution of the nonhomogeneous linear ODE 
y + py — r. 



Using a method of this section or separating variables, find 
the general solution. If an initial condition is given, find 
also the particular solution and sketch or graph it. 

22. / + y = y 2 , y(0) = -§ 

23. y + xy = xy -1 , v(0) = 3 

24. y + y = —x/y 

25. y = 3.2y - 10y 2 

26. y = (tan y)/{x - 1), y(0) = j7T 

27. y = \/{6e y - 2x) 

28. 2xyy' + (x — l)y 2 = x 2 e x (Sety 2 = z) 

29. REPORT PROJECT. Transformation of ODEs. 

We have transformed ODEs to separable form, to exact 
form, and to linear form. The purpose of such 
transformations is an extension of solution methods to 
larger classes of ODEs. Describe the key idea of each 
of these transformations and give three typical exam- 
ples of your choice for each transformation. Show each 
step (not just the transformed ODE). 

30. TEAM PROJECT. Riccati Equation. Clairaut 
Equation. Singular Solution. 

A Riccati equation is of the form 

(14) y + p(x)y = g(x)y 2 + h{x). 

A Clairaut equation is of the form 

(15) y — xy' + g(y'). 

(a) Apply the transformation y = Y + l/u to the 
Riccati equation (14), where Tis a solution of (14), and 
obtain for u the linear ODE u + (2 Yg — p)u = —g. 
Explain the effect of the transformation by writing it 
as y = Y + v, v = l/u. 

(b) Show that y = Y = x is a solution of the ODE 
y — (2x 3 + 1 ) y = — x 2 y 2 — x 4 — x + 1 and solve this 
Riccati equation, showing the details. 

(c) Solve the Clairaut equation y' 2 — xy' + y = 0 as 
follows. Differentiate it with respect to x, obtaining 
y"(2 y - x) = 0. Then solve (A) y" = 0 and (B) 
2y — x = 0 separately and substitute the two solutions 
(a) and (b) of (A) and (B ) into the given ODE. Thus 
obtain (a) a general solution (straight lines) and (b) a 
parabola for which those lines (a) are tangents (Fig. 6 
in Prob. Set 1.1); so (b) is the envelope of (a). Such a 
solution (b) that cannot be obtained from a general 
solution is called a singular solution. 

(d) Show that the Clairaut equation (15) has as 
solutions a family of straight lines y = cx + g{c) and 
a singular solution determined by g'(s) — —x, where 
s = y , that forms the envelope of that family. 



31. Newton’s law of cooling. If the temperature of a cake 
is 300°F when it leaves the oven and is 200°F ten 
minutes later, when will it be practically equal to the 
room temperature of 60°F, say, when will it be 61°F? 

32. Heating and cooling of a building. Heating and 
cooling of a building can be modeled by the ODE 

T' = k,(T - T a ) + k 2 (T - TJ + P, 

where T = T(t) is the temperature in the building at 
time ?, T a the outside temperature, T w the temperature 
wanted in the building, and P the rate of increase of T 
due to machines and people in the building, and kj and 
k 2 are (negative) constants. Solve this ODE, assuming 
P = const, T w = const, and T a varying sinusoidally 
over 24 hours, say, T a = A — Ccos(27r/24)t.Discuss 
the effect of each term of the equation on the solution. 

33. Drug injection. Find and solve the model for drug 
injection into the bloodstream if, beginning at t = 0, a 
constant amount A g/min is injected and the drug is 
simultaneously removed at a rate proportional to the 
amount of the drug present at time t. 

34. Epidemics. A model for the spread of contagious 
diseases is obtained by assuming that the rate of spread 
is proportional to the number of contacts between 
infected and noninfected persons, who are assumed to 
move freely among each other. Set up the model. Find 
the equilibrium solutions and indicate their stability or 
instability. Solve the ODE. Find the limit of the 
proportion of infected persons as t — * 00 and explain 
what it means. 

35. Lake Erie. Lake Erie has a water volume of about 
450 km 3 and a flow rate (in and out) of about 175 km 2 


CHAP. 1 First-Order ODEs 

per year. If at some instant the lake has pollution 
concentration p = 0.04%, how long, approximately, 
will it take to decrease it to p/2, assuming that the 
inflow is much cleaner, say, it has pollution 
concentration p/4, and the mixture is uniform (an 
assumption that is only imperfectly true)? First guess. 

36. Harvesting renewable resources. Fishing. Suppose 
that the population y(t) of a certain kind of fish is given 
by the logistic equation (11), and fish are caught at a 
rate Hy proportional to y. Solve this so-called Schaefer 
model. Find the equilibrium solutions Vi and y 2 (> 0) 
when H < A. The expression Y = Hy z is called 
the equilibrium harvest or sustainable yield corre- 
sponding to H. Why? 

37. Harvesting. In Prob. 36 find and graph the solution 
satisfying y(0) = 2 when (for simplicity) A = B = 1 
and H = 0.2. What is the limit? What does it mean? 
What if there were no fishing? 

38. Intermittent harvesting. In Prob. 36 assume that you 
fish for 3 years, then fishing is banned for the next 
3 years. Thereafter you start again. And so on. This is 
called intermittent harvesting. Describe qualitatively 
how the population will develop if intermitting is 
continued periodically. Find and graph the solution for 
the first 9 years, assuming that A = B = 1, H = 0.2, 
and y(0) = 2. 

Fig. 23. Fish population in Problem 38 

39. Extinction vs. unlimited growth. If in a population 
y (t) the death rate is proportional to the population, and 
the birth rate is proportional to the chance encounters 
of meeting mates for reproduction, what will the model 
be? Without solving, find out what will eventually 
happen to a small initial population. To a large one. 
Then solve the model. 

40. Air circulation. In a room containing 20,000 ft 3 of air, 
600 ft 3 of fresh air flows in per minute, and the mixture 
(made practically uniform by circulating fans) is 
exhausted at a rate of 600 cubic feet per minute (cfm). 
What is the amount of fresh air y(t) at any time if 
y(0) = 0? After what time will 90% of the air be fresh? 

1.6 Orthogonal Trajectories. Optional 

An important type of problem in physics or geometry is to find a family of curves that 
intersects a given family of curves at right angles. The new curves are called orthogonal 
trajectories of the given curves (and conversely). Examples are curves of equal 
temperature (isotherms) and curves of heat flow, curves of equal altitude (contour lines) 
on a map and curves of steepest descent on that map, curves of equal potential 
(equipotential curves, curves of equal voltage — the ellipses in Fig. 24) and curves of 
electric force (the parabolas in Fig. 24). 

Here the angle of intersection between two curves is defined to be the angle between 
the tangents of the curves at the intersection point. Orthogonal is another word for 

In many cases orthogonal trajectories can be found using ODEs. In general, if we 
consider G(x, y, c) = 0 to be a given family of curves in the xy-plane, then each value of 
c gives a particular curve. Since c is one parameter, such a family is called a one- 
parameter family of curves. 

In detail, let us explain this method by a family of ellipses 


(c > 0) 

SEC. 1.6 Orthogonal Trajectories. Optional 


and illustrated in Fig. 24. We assume that this family of ellipses represents electric 
equipotential curves between the two black ellipses (equipotential surfaces between two 
elliptic cylinders in space, of which Fig. 24 shows a cross-section). We seek the 
orthogonal trajectories, the curves of electric force. Equation (1 ) is a one-parameter family 
with parameter c. Each value of c (> 0) corresponds to one of these ellipses. 

Step 1. Find an ODE for which the given family is a general solution. Of course, this 
ODE must no longer contain the parameter c. Differentiating (1), we have x + 2yy = 0. 
Hence the ODE of the given curves is 

(2) y =f(x,y) = - J *-. 

2 y 

Fig. 24. Electrostatic field between two ellipses (elliptic cylinders in space): 
Elliptic equipotential curves (equipotential surfaces) and orthogonal 
trajectories (parabolas) 

Step 2. Find an ODE for the orthogonal trajectories y = y(x). This ODE is 

( 3 ) 



fix, y) 

= + 

2 y 


with the same /as in (2). Why? Well, a given curve passing through a point (x 0 > y 0 ) has 
slope /(x 0 , To) at that point, by (2). The trajectory through (x 0 , y 0 ) has slope — 1 //(jc 0 , To) 
by (3). The product of these slopes is —1, as we see. From calculus it is known that this 
is the condition for orthogonality (perpendicularity) of two straight lines (the tangents at 
(x 0 , .Vo)), hence of the curve and its orthogonal trajectory at (x 0 , y 0 ). 

Step 3. Solve (3) by separating variables, integrating, and taking exponents: 

dy -Ci be .|~i - . ~ *2 

— = 2 — , In \y | = 2 In x + c, y = c x . 

y x 

This is the family of orthogonal trajectories, the quadratic parabolas along which electrons 
or other charged particles (of very small mass) would move in the electric field between 
the black ellipses (elliptic cylinders). 


CHAP. 1 First-Order ODEs 




Represent the given family of curves in the form 
G(x, y; c) = 0 and sketch some of the curves. 

1. All ellipses with foci —3 and 3 on the x-axis. 

2. All circles with centers on the cubic parabola y = x 3 
and passing through the origin (0, 0). 

3. The catenaries obtained by translating the catenary 
y — cosh x in the direction of the straight line y = x. 


Sketch or graph some of the given curves. Guess what their 
OTs may look like. Find these OTs. 

4. y = x 2 + c 5. y = cx 

6. xy = c 7. y = c/x 2 

8. y = Vx + c 9. y = ce~ x 

10 . x 2 + (y - cf = c 2 


11. Electric field. Let the electric equipotential lines 

(curves of constant potential) between two concentric 
cylinders with the z-axis in space be given by 
u(x, y) = x 2 + y 2 = c (these are circular cylinders in 
the xyz-space). Using the method in the text, find their 
orthogonal trajectories (the curves of electric force). 

12. Electric field. The lines of electric force of two opposite 
charges of the same strength at ( — 1,0) and (1,0) are 
the circles through ( — 1, 0)and (1,0). Show that these 
circles are given by x 2 + (y — c) 2 = 1 + c 2 . Show 
that the equipotential lines (which are orthogonal 
trajectories of those circles) are the circles given by 
(x + c*) 2 + y 2 = c* 2 — 1 (dashed in Fig. 25). 

Fig. 25. Electric field in Problem 12 

13. Temperature field. Let the isotherms (curves of 
constant temperature) in a body in the upper half-plane 
y > 0 be given by 4x 2 + 9y 2 = c. Find the ortho- 
gonal trajectories (the curves along which heat will 
flow in regions filled with heat-conducting material and 
free of heat sources or heat sinks). 

14. Conic sections. Find the conditions under which 
the orthogonal trajectories of families of ellipses 
x 2 /a 2 + y 2 /b 2 = c are again conic sections. Illustrate 
your result graphically by sketches or by using your 
CAS. What happens if a -> 0? If b -> 0? 

15. Cauchy-Riemann equations. Show that for a family 
u(x, y) = c = const the orthogonal trajectories v(x, y) = 
c* = const can be obtained from the following 
Cauchy-Riemann equations (which are basic in 
complex analysis in Chap. 13) and use them to find the 
orthogonal trajectories of e x sin y = const. (Here, sub- 
scripts denote partial derivatives.) 

U x Vy , Uy V x 

16. Congruent OTs. Ify r =/(x) with /independent of y, 
show that the curves of the corresponding family are 
congruent, and so are their OTs. 

Existence and Uniqueness of Solutions 
for Initial Value Problems 

The initial value problem 

|y'| + W=0, y(0) = 1 

has no solution because y = 0 (that is, y(x ) = 0 for all x) is the only solution of the ODE. 
The initial value problem 

y' = 2x, 

y( 0) = l 

SEC. 1.7 Existence and Uniqueness of Solutions 


has precisely one solution, namely, y = x 2 + 1. The initial value problem 

xy' = y - l, y(0) = l 

has infinitely many solutions, namely, y = 1 + cx, where c is an arbitrary constant because 
y(0) = 1 for all c. 

From these examples we see that an initial value problem 

(1) y = f(x, y), y{xf) = y 0 

may have no solution, precisely one solution, or more than one solution. This fact leads 
to the following two fundamental questions. 

Problem of Existence 

Under what conditions does an initial value problem of the form (1) have at least 
one solution ( hence one or several solutions)? 

Problem of Uniqueness 

Under what conditions does that problem have at most one solution ( hence excluding 
the case that is has more than one solution)? 

Theorems that state such conditions are called existence theorems and uniqueness 
theorems, respectively. 

Of course, for our simple examples, we need no theorems because we can solve these 
examples by inspection; however, for complicated ODEs such theorems may be of 
considerable practical importance. Even when you are sure that your physical or other 
system behaves uniquely, occasionally your model may be oversimplified and may not 
give a faithful picture of reality. 


Existence Theorem 

Let the right side f(x, v) of the ODE in the initial value problem 

(1) y' =f(x,y), y(x 0 ) = yo 

be continuous at all points ( x , y) in some rectangle 

R: \x - x 0 \ < a, |y - y 0 | < b (Fig. 26) 

and bounded in R; that is, there is a number K such that 

(2) | f(x, y)\ = K for all (x, v) in R. 

Then the initial value problem (1) has at least one solution y(x). This solution exists 
at least for all x in the subinterval \x — jtol < a of the interval \x — xol < 
here, a is the smaller of the two numbers a and b/K. 


CHAP. 1 First-Order ODEs 







Fig. 26. Rectangle R in the existence and uniqueness theorems 

( Example of Boundedness. The function /(x, y) = x 2 + y 2 is bounded (with K = 2) in the 
square |x| < 1 , |_y | < 1. The function f(x, y) = tan (x + y) is not bounded for 
|x + y| < 77/2. Explain!) 


Uniqueness Theorem 

Let f and its partial derivative f y = df/dy be continuous for all (x, y) in the rectangle 
R (Fig. 26) and bounded, say, 

(3) (a) |/(x, y) | ^ K, (b) \f y (x,y)\ ^ M for all (x, y) in R. 

Then the initial value problem (1) has at most one solution y(x). Thus, by Theorem 1, 
the problem has precisely one solution. This solution exists at least for all x in that 
subinterval \x — XqI < a. 

Understanding These Theorems 

These two theorems take care of almost all practical cases. Theorem 1 says that if f(x, y) 
is continuous in some region in the xv-plane containing the point (xo, VoX then the initial 
value problem ( 1 ) has at least one solution. 

Theorem 2 says that if, moreover, the partial derivative df/ dy of / with respect to y 
exists and is continuous in that region, then (1) can have at most one solution; hence, by 
Theorem 1, it has precisely one solution. 

Read again what you have just read — these are entirely new ideas in our discussion. 

Proofs of these theorems are beyond the level of this book (see Ref. [All] in App. 1); 
however, the following remarks and examples may help you to a good understanding of 
the theorems. 

Since y' = f(x,y), the condition (2) implies that \y' Si K\ that is, the slope of any 
solution curve y(x) in R is at least —K and at most K. Hence a solution curve that passes 
through the point (xo, yo) must lie in the colored region in Fig. 27 bounded by the lines 
1 1 and 1 2 whose slopes are —K and K, respectively. Depending on the form of R, two 
different cases may arise. In the first case, shown in Fig. 27a, we have b/K a and 
therefore a = a in the existence theorem, which then asserts that the solution exists for all 
x between xo — a and xo + a. In the second case, shown in Fig. 27b, we have b/K < a. 
Therefore, a = b/K < a, and all we can conclude from the theorems is that the solution 

SEC. 1.7 Existence and Uniqueness of Solutions 



exists for all x between x 0 — b/K and xo + b/K. For larger or smaller x’s the solution 
curve may leave the rectangle R, and since nothing is assumed about / outside R , nothing 
can be concluded about the solution for those larger or amaller x’s; that is, for such x’s 
the solution may or may not exist — we don’t know. 

(a) ( 6 ) 

Fig. 27. The condition (2) of the existence theorem, (a) First case, (b) Second case 

Let us illustrate our discussion with a simple example. We shall see that our choice of 
a rectangle R with a large base (a long x-interval) will lead to the case in Fig. 27b. 

Choice of a Rectangle 

Consider the initial value problem 

y' = 1 + v 2 , y(0) = 0 

and take the rectangle R: \x\ < 5, |y| < 3. Then a = 5, b = 3. and 




= |l + y 2 | £ K = 10, 
= 2|y| £ M = 6, 


< a. 

Indeed, the solution of the problem is y = tan* (see Sec. 1.3, Example 1). This solution is discontinuous at 
±77/2, and there is no continuous solution valid in the entire interval |*| < 5 from which we started. 

The conditions in the two theorems are sufficient conditions rather than necessary ones, 
and can be lessened. In particular, by the mean value theorem of differential calculus we 

fix, y 2 ) - f(x, >’ 1 ) 

(*2 “ Ti) 




where (x, yi) and (x, y 2 ) are assumed to be in R, and y is a suitable value between yi 
and y 2 - From this and (3b) it follows that 

(4) I/O, y 2 ) ~ fix, Vi)| S M\y 2 — ^il -